I'd approach this using a custom shader on my SpriteRenderers that takes two textures, one for each version you want to transition between.
(Even if using this effect to transition between more than two states, I'd still try to structure the game so we only need two for rendering at any one time: the old state and the next state. We'll let that transition play to completion before we let the player initiate a transition to a third state, at which point we can replace our previous "old" state since it's no longer visible)
This shader would cross-fade between the two textures, using a third single-channel texture as a mask for the effect (eg. black = 100% old state, grey = 50/50 blend, white = 100% new state). I'd map this mask texture in screenspace.
Then I'd set up a second camera to render to a RenderTexture, to generate this mask. This second camera would move with my main camera and match its FoV/orthographic size, so any animation in my view applies to both the world and reveal mask consistently. This camera would render before my main camera, and be set up to see only content on a specially-designated "reveal" layer. In the video example, it looks like they use a particle system to splat-out a collection of little cloud shapes into this reveal texture, which expand over time, giving the transition an organic-looking spreading edge.