The exact evolution sequence
Learn an attack, upgrade it to its maximum level, and take the required stat card at least once during that same run. When an evolution is ready, the level-up offer inserts an evolution card. Select that gold card to evolve; meeting the conditions alone does not transform the move.
Why a move will not evolve
Check four things: the move has an evolution, it is at its own maximum level, you took the matching run stat card, and it has not already evolved. Buff moves have no evolution. A permanent upgrade of the same name does not count as a run card.
Move evolution is not species evolution
Ember becomes Fire Blast, but Charmander does not automatically become Charmeleon. Partner selection and move progression are different systems. Use the checker to identify the next missing action for your current move.