An evolution is not automatic. You need two things at once: the base move at max level, and the specific upgrade card that move asks for. When both are true, a golden evolution card enters the level-up pool and converts your move into its stronger form.
There are 59 evolvable moves out of 64. Seven different upgrade cards serve as the key — Power, Vigor, Reflexes, Sprint, Armor, Synthesis and Magnet — and which one you need depends on the move, not on its type.
Power
The most common key. It unlocks evolutions for straightforward damage moves such as Ember, Water Gun, Vine Whip and Rock Throw.
Vigor
Pairs with moves that trade speed for weight, like Tackle becoming Take Down or Body Slam becoming Giga Impact.
Reflexes
Feeds the evolutions that add shots, jumps or pierce, such as Hyper Voice into Uproar or Thunderbolt into Zap Cannon.
Sprint
Drives movement-flavoured moves: Quick Attack to Extreme Speed, Flame Wheel to Flare Blitz, Bubble to Surf.
Armor, Synthesis and Magnet
The rarer keys. Armor covers turrets and fields, Synthesis covers orbits and trails, Magnet covers the psychic and ghost evolutions.
Four moves never evolve
Swords Dance, Agility, Harden and Aqua Ring deal no damage at all: they are stat engines, and they stay as they are.