The question is discussed in detail in the paper "Families of group actions, generic isotriviality, and linearization" (Transform. Groups 19 (2014), no. 3, 779–792. MR 3233525) by Peter Russell and myself. There is e.g. an example, due to Nori, of a flat morphism $X \to Y$ with fibers $A^1$ and base $Y$ equal the cuspidal curve which is not locally trivial in the etale topology.
