This first part of dialog happens the first time you talk to them after beating the Magus in this era, so you can see it much earlier.

Fiona: My husband Marco had gone
as a soldier to fight Magus's army,
but he has returned to me....
I, I....

Marco: Relax.
From now on, we'll be together for good.

MARCO: Don't worry.
I'm home for good.

The sidequest appears as soon as you tell the woman with the seedling in B.C.12000 to save it.

Fiona: No matter what I plant,
it soon withers....
It's the doing of monsters that have
settled in somewhere in the desert.
If I were to plant this seedling with mysterious
powers that was passed down for generations,
it could even restore this desert to green earth....
But I can't plant it with monsters nesting
in the desert cave....

Marco: It seems the monsters nesting in
the desert cave are weak against water.
But with my strength, trying to beat those
monsters would be....

Strange Happening plays here. The entrance resembles a whirlpool of sand, and the sand inside the caves moves in irregular currents. Things get more interesting deeper inside when the ground starts shaking.

A strange skeletal something sticks partway out of the ground, then sinks back under the sands.

Marle: It's that monster's fault the
forest is dying out so fast!

{Marle}: So that's the monster
responsible for this desert!

Lucca: It looks like that monster is
the source of the desertization.

{Lucca}: The monsters are creating a
desert here.

Frog: Th, that thing turned the forest
into desert!

{Frog}: So, THEY changeth the forest
into the desert!

Robo: The progressing desertization
IS that monster's fault.

{Robo}: The monsters are causing
the desertification here.

Magus: Melfikk....
So that's the cause of the desertization....

Two 'k's make it sound twice as sinister.

Marle: But something like a monster that
moves around underground....
You can't even guess where to run into it!

{Marle}: But how can one find a
monster that roams the inner earth?

Lucca: But a monster that moves
underground is a nuisance.

{Lucca}: It will be hard to defeat a
monster that travels underground!

Frog: But for a lowlife that moves
underground to be the opponent....
This guy's a pain.

{Frog}: How to slayeth monsters
within the earth...
'Tis a very difficult task.

Robo: HOWEVER, for the monster to
be able to come and go underground....

{Robo}: But monsters that travel
underground are impossible to find!

Ayla: Bones moving around underground!
Big trouble to find!

{Ayla}: Hard to find monster in
earth!

Catching the monster when it's above ground starts combat. Though Water attacks inflict no damage, they harden the sandy bones and drop the defense of the top and bottom sections. Other attacks will restore their defense, but only if the core is dead. To make the battle easier, keep the core alive by healing it with any element except Water. This will also limit the attacks the boss can use. Rob the core for a Speed Capsule.

Marle: Now the trees won't die off any
more either. Fiona-san will be glad.

Lucca: Now the cause of the desertization
is removed. We ought to tell Fiona-san.

{Lucca}: We've neutralized the
source of this desert.
We should tell Fiona.

Frog: Now there's nothing left to worry about.
Let's let Fiona know.

Robo: The progress of the desertization
SHOULD now SUBSIDE.
LET US notify Fiona-san.

Ayla: Bones that break trees dead.
Fiona glad!

Marco: I can't believe you exterminated
those monsters for us!
How can we thank you...?

MARCO: You killed the monsters?!
How can I ever thank you?

Fiona: Thank you for exterminating
the monsters for us.
I will try doing all I can once more.
But, to make the desert into forest
once again, we will need an
unimaginably long time....
Ahh....
If only there was someone who could
keep working for hundreds of years, we
could turn this desert back to forest....

Robo: ......
Well, EVERYONE?
I shall remain HERE and act
as Fiona-san's HELPER.
You can just COLLECT ME after
THIS desert IS revived as forest.
May I remain HERE and act as
HELPER?

Robo goes out into the desert, and can from now on be seen performing various tasks related to growing the forest.

The area, which was previously almost barren, is now covered with forest, and this building stands in the middle. There are several nuns, priestesses, or whatever you want to call them inside.

This is a bit of a loose translation. 神体 (shintai) actually refers to an object in which the spirit of a deity resides, and the line seems to indicate that they consider Robo to be an inanimate object that was animated by a god.

She sells various hats with protection from status ailments, including the Safeguard Hat. They're expensive, but are some of the strongest headgear, and anyone can wear them. Robo sits inert on a pedestal in the back, looking worn down and rusty. They move him to the ground and he starts moving.

Robo: WH, WHERE...?
OH....
OHH....
{char1}....
I... I MISSED YOU....
NO... to YOU, it WAS a matter
of an INSTANT.
HOWEVER, for ME, 400 years
WAS a long time....
HOWEVER, HARDSHIPS bear results,
and the forest HAS BEEN revived.
......
NOW, tonight, SHALL WE NOT
celebrate our 400-year REUNION?

That night, everyone relaxes around a campfire in the forest. The Farthest Reaches of Time plays. Lucca continues repairing Robo as he speaks.

Robo: Having now taken a journey of
400 years, I HAVE realized something.
WE HAD thought that the Gates appeared
due to the distortions from Lavos's power,
but the feeling HAS come upon me that we
were mistaken.

{Robo}: After 400 years of
experience, I have come to think that
Lavos may not be responsible for the
Gates.

Robo: I cannot be CONFIDENT about it,
but did someone not perhaps want
to show us something...?
Something through the Gates in various eras.
Or, MAYBE that someone wanted to see it
personally.
In order to think back upon the way his or
her own life went....

{Robo}: I have come to think that
someone, or something wanted us to
see all this.
The different events over time, that
we have witnessed.
It is almost as if some entity wanted
to relive its past.

Ayla: Ayla understand that.
Legend that people see all
memories up to then when die!

{Ayla}: {Ayla} know!
When people die, elders say, see
whole life pass by!

Frog: They say when you die, the memories
etched deeply into your heart while you
lived come to mind one after another.
Some are happy memories too, but
generally they're sorrowful memories.

{Frog}: 'Tis true that mortals do
relive their most profound memories
before death claimeth them.
Yet those memories most often are
sad ones.

Robo: Undoubtedly,
"I want to go back to that time,"
"If I had just done like so then,"
...strong feelings such as these
LIKELY call memories to mind.

{Robo}: Thinking things like, "If only
I had done this," or, "I shouldn't
have done that..." triggers
unpleasant, old memories.

Marle: Do you have one, Lucca?
A moment you want to return to?

{Marle}: Is there a point in time you'd
want to return to, {Lucca}?

{Marle}: I'm sorry, was that
something I shouldn't have asked?

Lucca: I try to think about it as
little as possible.
After all, it just wears me out to.

Frog: Anyway....
The owner of these memories is
really tangled up with Lavos.
Every one of these eras involves Lavos.

{Frog}: Lavos playeth an integral
role in the fortunes of this Entity...

Magus: ......
So, who are you saying it is?

Robo: I do NOT know whose
memories they are.
Perhaps not a person....
MAYBE a larger existence.
The day we understand that MAY
come when OUR journey ends.
...SHALL we get to sleep then?

Who is this mysterious entity? The going fan theory seems to be the planet itself, which fits quite well actually.

The music fades and everyone goes to sleep, but Lucca wakes up in the middle of the night. There's a faint background noise similar to the sound of travel through a Gate. Lucca finds a Gate in the nearby forest and enters. It's red when it opens instead of the usual blue. The Gate opens into a room, the sounds get louder, and Lucca looks confused. The room looks like Lucca's, but lacks the usual clutter of books, machines, and chemicals.

There's a note on the ground.

"6/24, A.D.990
Dad promised to go hiking, but he
skipped out on it to do research.
Lucca doesn't get it at all....
But that's fine. Lucca's gonna be a
normal bride, so she doesn't gotta
know about science and stuff!

This appears to be from a diary written by Lucca ten years in the past. The third person is an indication of childishness.

The kitchen downstairs has another note on the table.

"The passcode to stop the machine is....
My most beloved person.
Enter this in an emergency.
　　　　　　　　　　　　　　　　Taban"

As soon as Lucca enters the main room of the house, a scene starts and the sounds get even louder. Lucca's mother Lara is in the room along with a young purple-haired girl and a large machine. The others don't notice Lucca, who looks shocked.

Lara: Dear me, what's this machine...?
Taban says not to get close to it because
it's dangerous, but he's gotten it so dirty.

She walks to one side of the contraption, where it looks like a conveyor belt, then seems surprised.

Lara: Huh?
The hem of my skirt got caught in it.
Lucca, oh, Lucca, help me out a bit.

The little girl, who must be the Lucca of ten years past, goes over to help.

With a clicking sound, the machine starts to move, dragging Lara in.

Lara: !!
Lucca! Stop the machine!
Enter the passcode.

Lucca: I don't get it!
Mom!

Young Lucca hurries to the controls, but just runs back and forth helplessly. However, the older Lucca is in a position to change the past....

Lara: Ahh, it's no good....
Darling!!

The scene fades out, the sounds stop, and there's a scream. Fade in to (older) Lucca collapsed at the foot of her bed. There are now two notes in the room, blocking every exit, except for the Gate, which has reappeared.

"7/1, A.D.990
I heard dad and the
doctor talking.
But I wish I hadn't....
Mom... her legs....

"7/15, A.D.990
If only I knew all about machines,
nothing like this would have happened....
If only I....

Note that this entry is written in first person. The shift indicates a movement from childish innocence to adult maturity.

The Gate returns Lucca to the present. Robo stands in the clearing. Everyone else is still sleeping.

Robo: I WILL give her MY legs.
They ARE well-balanced ultra-
miniaturized bipedal walking parts.
If you would just attach caterpillars
or something to ME....
PLEASE do not be depressed.

{Robo}: I'll give you my legs!
They're lightweight and well
balanced.
You could mount some treads on me
instead.

Lucca: Thanks.
I have such a good friend....

{Lucca}: Thanks, you're such a good
friend.

Robo: Friend....
I, a robot....

{Robo}: A...friend?
Me, a robot?
You called me your friend?

The code in the Japanese version is L A L A, not L A R A. I'm fairly certain that this agrees with the official system of Romanizing Japanese, which sees little actual use. But Lala makes a lousy name in English, so they changed it for the NA release. It's also possible that the ambiguity of how to Romanize her name was done deliberately to make it more difficult to get the code correct. Text from the prerelease suggests it was originally outright impossible.

Young Lucca runs up to her mother.

Fade in to Lucca collapsed at the foot of her bed. There are now two notes in the room, blocking every exit except for the Gate, which has reappeared, though one of the notes is just the one that was here when Lucca arrived.

"7/2, A.D. 990
Wow, thank goodness. I'm thinking I'll learn
a little more about machines so I'll be okay
in case something like that happens again.

It's unclear whether this entry is in first or third person, which may mean she grew up a little, but not as abruptly as in the scenario with the tragedy.

The Gate returns Lucca to the present. Robo stands in the clearing. Everyone else is still sleeping.

Robo: It HAS BEEN weighing
on your mind the whole time.
You ARE a kind person, Lucca.

The scene ends the same way regardless of which branch was taken. Robo slides open some of his plating, revealing something shiny inside.

Robo: I will give you a present.
I MADE it by hardening the RESIN
of trees that grew in the forest.
This jewel HAS four hundred years of meaning.
Please make some use of it.

Lucca: Thanks...
Robo.

{Lucca}: {Robo}...
you're so sweet.

Thus concludes the sidequest. Rewards: Speed Capsule, Dream of Greenery, the hat shop, and the infamous "Entity" discussion.