The new trailer highlights a selection of attacks from Azumarill's moveset, though does not explicitly give names to any of the moves shown. Based on what was already known about the Pokémon from earlier leaks, the attacks shown off appear to be Whirlpool, which deals area-of-effect damage while restoring Azumarill's HP; Play Rough, which increases Azumarill's move speed and changes Azumarill's next basic attack to a lunging attack that deals repeated blows and stuns; and Aqua Tail, which gives a variety of buffs to Azumarill's next three base attacks. You can find more detail on the effects of these moves on Bulbapedia.
The official Pokémon UNITE Twitter page has also teased the coming addition of a new Holowear set. Although it is obscured, it appears to be a spring-themed set for Wigglytuff.