The Rise of Darkrai
Palkia was battling against its enemy, Dialga. During the battle, Dialga used Roar of Time against it and injured its left arm. Palkia then fleed to its own dimension in Alamos Town. Due to Palkia's arrival, the Pokémon who were having nightmares under the influence of Darkrai's Dark Void woke up. Simultaneously, Palkia's presence causes the dimension Dialga and Palkia were fighting in to disintergrate the normal dimension little by little. Dialga, who was furious at Palkia for its cowardice, engaged it to continue the battle which did enormous damage to the entire town. During the battle, Darkrai tried to hold them off as long as he could, but ended up getting himself killed in the process of saving the town. Before Dialga and Palkia can deliver the final blow against each other, Ash and Dawn successfully managed to magnify the volume of the song, Oración, soothing both Dialga and Palkia in mid-flight and healing Palkia's wound. Just as Dialga returned to its dimension, Ash and Dawn asked Palkia to return Alamos Town back to its original dimension. Once everything was restored, Palkia returned to its dimension.
Giratina and the Sky Warrior
Palkia didn't make an appearance but was mentioned in a flashback of its battle in Alamos Town. However, its counterpart, Dialga, played a minor role in the beginning of the movie.
Arceus and the Jewel of Life
Palkia arrived in Michina Town to rescue Dialga from being sucked into a deadly spatial distortion by using Spacial Rend to close it. Sheena then connected with the hearts of both Dialga and Palkia and puts an end to their conflict with each other once and for all. After that, Dialga and Palkia returned to their respective dimensions. Sheena and Kevin explained to Ash and his friends that the reason why Dialga and Palkia clashed in Alamos Town was because both of their dimensions, which were not meant to ever intersect, intersected due to Arceus's preparation to awaken from its long, eternal slumber. As a result, both Dialga and Palkia believed that they both invaded each other's territory and the battle also affected the Reverse World which made it bring Giratina into the conflict as well. When Arceus arrived and prepared to destroy all of the humans due to Damos' betrayal, Dialga, Palkia, and Giratina arrived and protected Ash and his friends from Arceus's attacks. In order to by Dialga some time to send Ash and his friends back in the past to change the timeline and save the future, Palkia used is space bending abilities to keep Arceus at bay. However, it ended up getting severely wounded by the angered Alpha Pokémon along with Dialga and Giratina. Eventually, when Ash and the others had successfully changed the timeline for the better, Arceus healed Palkia, Dialga, and Giratina, and Palkia returned to its own dimension to live peacefully once more.
Palkia, despite being a powerful Pokémon able to bend space at it's will, don't seem so addicted to battling. It have a far more clear sense of strategy than it's temporal counterpart, having a variety of moves that prevent each other's weaknesses. It's Aura Sphere is good against it's half-weakness, the Ice type, and the Steel type that resists it's signature move, Spacial Rend. Also, it's coincidentally useful against Dialga and Darkrai, both enemies that he never met before and fought with in Alamos town. Plus, Palkia rarely uses it's space-bending skills to battle, using it in The Rise of Darkrai to hide itself from Dialga and to contain Arceus in Arceus and The Jewel of Life, possibly in a desperate attempt to gain more time. It seems more self-defensive, fighting only for it's territory in The Rise of The Darkrai and to protect itself from Dialga on the same movie.
Yet, Palkia also shows quite some knowledge about battling and Pokémon moves, as show when it didn't tried to dodge Darkrai's undodgeable Shock Wave, instantly using Protect. It also showed an equal amount of knowledge about Pokémon type advantages, as shown when it prioritized it's Aura Sphere, a fighting-type move, against Dialga and Darkrai, respectively Steel and Dark-type Pokémon.
There's no need to say that it's very territorial, just like it's counterpart Dialga, engaging in a serious battle once it's territory was violated. Despite it's respectable battling skills, it seemed to be way more cautious than Dialga, since it used the smoke generated by a direct Aura Sphere to retreat once it was hurt. In this point, it was shown that Palkia could have possibly developed some kind of fear from Dialga, given it flew over the city with Dialga in pursuit instead if fighting once it got cornered. Of course, it's just a possibility.
Palkia was shown to be willing to forgive, not engaging any other battle with Dialga since the events of The Rise of Darkrai. Also, it came to Dialga's rescue in Arceus and The Jewel of Life, even without being asked to. It's not know if Palkia restored Darkrai back to life accidentally or purposely, so we're not able to extract any possible personality trait from that act.
Palkia can be seen as very brave when it's protecting someone. He fought Arceus almost to death in Arceus and The Jewel of Life to protect humanity from Arceus's mistaken wrath, despite having full knowledge that Arceus was way more Powerful than it. Strangely, Palkia, nor Dialga or Giratina for that matter, were seen using it's signature moves or any other Dragon-type moves against Arceus, despite being stab and the unique moves Arceus couldn't be immune against, given the Fairy typing didn't exist at the time. Also, when Arceus got immunity against it's Aura Sphere and Hyper Beam at once, Palkia strategically decided to slow Arceus down with it's space-bending abilities.
One curiosity about Palkia's and Dialga's relation is that, differently from the Palkia and Dialga shown in The Battle Finale of Legend (Pokémon DP Series Episode), these ones hadn't met before The Rise of Darkrai and were never supposed to meet.
|Spacial Rend||The Rise of Darkrai|
|Aura Sphere||The Rise of Darkrai|
|Hyper Beam||The Rise of Darkrai|
|Protect||The Rise of Darkrai|
|+ indicates this Pokémon used this move recently.*|
- indicates this Pokémon normally can't use this move.