Ways to show two structures are elementary equivalent

Question

Let $\mathcal{L}$ be a finite first-order language. When we say structure we mean $\mathcal{L}$-structure.

Question. Can someone lists different ways which we may use to show two given structures are elementarily equivalent?

For example:

1) doing induction on complexity of formulas,

2) using Ehrenfeucht-Fraisse games,

3) ....

Answer 1

3. Finding a theory satisfied by both structures and then proving that this theory is complete, for example by Vaught's test.

4. Building a forcing extension of the set-theoretic universe in which the structures become isomorphic. (This actually proves $L_{\infty,\omega}$-equivalence, and it tends to be similar to an argument about infinitely long Ehrenfeucht-Fraïssé games.)