I think that for getting an answer to this question you should provide a definition of what you call exactly a graph transformation, because I think that for the 'general case' there is no 'general approach'.

What I mean by graph transformation:

Let $G$ be a graph, we can define $H = (V,E)$ where $V$ and $E$ are define as functions of $G$.

Some examples would be the line graph (edge graph) and the dual graph.

For example, a precise definition of the line graph can be given as the following transformation of G, i.e., H is a line graph of G iff H = L(G), where I'm calling L a graph transformation.

Line graph of G is defined as H=(V,E) where

V = E(G)

E = \{ \{e_1,e_2\} : e_1,e_2\in E(G) and e_1\cap e_2 \neq \emptyset \}

where $e_1\cap e_2 \neq \emptyset$ means that e_1 and e_2 share at least one of their endpoints.

With that explained, my question is: Given a graph transformation F, what would be the best approach for proving (or disproving) that:

F(G_1) is isomorphic to F(G_2) iff G_1 is isomorphic to G_2.

I'm thinking in an induction based approach, but I'm not yet completely sure how I would do it. 

OK, I would try to find a counter example for the disproving part. Could you post which transformation exactly are you dealing with? I don't really think there is a general approach, anyway lets wait someone else to answer the question.

Here's a counter-example for the line graph transformation L:

Let G_1 be K_3 (i.e., the complete graph with 3 vertices) and G_2 be the complete bipartite graph K_{1,3}. G_1 and G_2 are not isomorphic, but their line graphs are isomorphic.

The Whitney graph isomorphism theory further states that except for this counter-example, G_1 is isomorphic to G_2 iff L(G_1) is isomorphic to L(G_2).