Probably what you really need is the Gromov-Phillips transversality theorem.

One could answer if the question were asked with precision (hypotheses and expected conclusion). If I understand correctly, the source manifold M has a foliation Fol; you consider a map F:MxS->N; and for each s the map F_s:M->N; and a submanifold R in N. You assume that F|(LxS) is transverse to R in N, for each leaf L. You ask if for almost every s and each leaf L, the map F_s|L is transverse to R in N. Is that it?