You’ll automatically receive this request the moment you enter Jubilife Village after completing the postgame mission “Seeking the Remaining Plates”. Ress the gatekeeper will mention seeing a shadow that looked like Giratina at the Cobalt Coastlands.
Where to Find Giratina
Naturally, if you’re looking for Giratina, you should depart for the Cobalt Coastlands. Besides that, there aren’t many clues for where Giratina could be, unless you’ve played the Diamond/Pearl/Platinum games or think outside the box. Essentially, there’s only one place that hasn’t served a purpose: Turnback Cave, near the north-west corner.
So make your way to Turnback Cave, either by traveling north from the Beachside Camp, ascending the cliffs with Sneasler, or gliding/swimming west from the Molten Arena. Use Sneasler to climb up the mountain at the center of Spring Path. In the center is a hollow with a cave at the west side. This should be all too familiar to some players.
How to Catch Giratina
Inside Turnback Cave, Giratina will be waiting patiently up ahead. Approach it to begin a battle.
This fight is essentially the same as the Volo battle, but all of your Pokémon will be healthy–and not worn out from Volo’s initial team. Therefore, it’ll be a much easier fight. That said, the focus here is on catching Giratina, not KOing it, so you may need to hold back somewhat.
As before, Giratina’s Shadow Force is the main thing to be cautious of. This attack not only hits hard, but makes Giratina obscured, boosting its evasion. Fielding a Normal-type (with access to non-Normal moves) will stop Giratina from using Shadow Force, but be wary as it has other strong moves too.
Fairy-types will take the full force of Shadow Force, but are immune to Giratina’s Dragon Claw and Dragon Pulse. What you could do is use Calm Mind and/or Baby-Doll Eyes to reduce the damage you take from Shadow Force. After Giratina runs out of Shadow Force (5 uses), it’ll be stuck with the very weak Earth Power.
When weakening Giratina, you can’t use False Swipe since it’s a Ghost-type. So you’ll have to gamble a bit to get Giratina’s HP low enough to capture. Don’t forget you can use moves in Agile Style to lower their damage.
How to Change Giratina’s Form
After capturing Giratina, you’ll obtain the Griseous Core. This key item can be used on Giratina to swap between its two forms: Altered and Origin. At the moment, Giratina will be in its more offensively-oriented Origin form (floating with no legs).