The parametric transversality theorem states that, given a parameterised family of smooth maps of $C^{\infty}$ manifolds $\phi_s:M \rightarrow N$ and a submanifold $R < N$ then for almost all values of the parameter $s$, $\phi_s$ is transverse to $R$ whenever the set of values of $s$ is a connected manifold and the adjoint map $F:M \times S \rightarrow N$, defined by $F(p,m)=\phi_s(p)$, is transverse to R. Here $S$ is the manifold of possible values of $s$.

The stronger property of a smooth map $\psi$ being transverse to a foliation $\mathcal{F}$ of $M$, is that $\psi$ is transverse to each leaf of the foliation.

Can we go one further and conclude that a family of maps smooth $\phi_s$ is transverse to a foliation of $M$ for almost all values of $s$? If not, under what conditions on $M,N, \psi_s$ and a given foliation would this hold?

I'm particularly interested in the case of a family of smooth maps from a real vector space to a compact Lie group. In my application, the foliation is into level sets of a given smoth function which is known to have only a single maximum.