A quite obvious, yet historically important example is represented by classical mechanics, in its Newtonian, Lagrangian and Hamiltonian presentations. In particular, the field was widely reorganized (and a lot was discovered) when it was realized that stationarity of the action can be assumed as a principle.

Of course the example is not perfect, because the level of generality of these theories is not the same, but considering only sufficiently regular motions it becomes close to what you search for.