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 could 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 buy Dialga some time to send Ash and his friends back in the past to change the timeline and save the future, Palkia used its 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 its will, does not seem so addicted to battling. It has a far more clear sense of strategy than its temporal counterpart, having a variety of moves that prevent each other's weaknesses. Its Aura Sphere is good against its half-weakness, Ice, and Steel, which resists its signature move, Spacial Rend. Also, it's coincidentally useful against Dialga and Darkrai, both enemies that it never met before and fought with in Alamos town. Plus, Palkia rarely uses its 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 its 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 shown when it had used Protect against Darkrai's undodgeable Shock Wave. It also showed an equal amount of knowledge about Pokémon type advantages, as shown when it prioritized Aura Sphere, a fighting-type move, against Dialga and Darkrai, who are respectively Steel and Dark-type Pokémon.
Needless to say, Palkia is very territorial, just like Dialga, engaging in serious battle once its territory was violated. Despite its respectable battling skills, it seemed significantly more cautious than Dialga, as it used the smoke generated by a direct Aura Sphere to retreat. At this point, it was shown that Palkia may have developed some substance of fear from Dialga, as it flew over the city with Dialga in pursuit rather than fighting when cornered. This seems somewhat understandable in The Rise of Darkrai, as Palkia was severely injured.
Palkia was shown to be willing to forgive, not engaging in any further battle with Dialga since the events of The Rise of Darkrai. The two counterparts fought alongside each other in Arceus and The Jewel of Life. Palkia also managed to totally restore Alamos Town and Darkrai in the end of The Rise of Darkrai.
Palkia can be seen as very brave when it protects someone. It fought Arceus to near death in Arceus and The Jewel of Life in an effort to protect humanity from Arceus' mistaken wrath, despite knowing that Arceus was notably more powerful. Strangely, neither Palkia, Dialga nor Giratina were seen using their signature moves or any Dragon-type moves on Arceus, despite the unique moves to which Arceus was not immune. Also, when Arceus held immunity against Palkia's Aura Sphere and Hyper Beam, Palkia strategically decided to slow Arceus down with its space-bending abilities.
One curiosity of Palkia's and Dialga's relation is that, different from the Palkia and Dialga shown in The Battle Finale of Legend (Pokémon DP Series Episode), the pair had not met before The Rise of Darkrai and were never supposed to meet.