Evolving mechanics involve a steadily advancing play style throughout the course of a single videogame. They often involve evolved player characters having some form of advantage or dominance over their previous forms.
Examples
Flow (the player controls advancing evolutions of a creature, each with a separate unique ability)
Space Station Silicon Valley (the player controls a microchip that can enter the bodies of various animals, each possessing its own ability, in a progression toward collecting the scattered remains of its robotic body, eventually re-assembling for a final battle)
Spore (the player develops a species from a microscopic organism through various levels of its evolution into an intelligent being that interacts with alien species across the galaxy. each phase of evolution exhibits its own play style, and the outcome of each affects the next.)
Worth Mention
Altered Beast (minimally)
Katamari series (although the player's perspective changes dramatically over time, as in Spore, the game's mechanics remain the same throughout)
Seaman (the mechanics of the game—which remain the player's responsibilities caring for the titular character throughout—change as the Seaman incrementally adapts)