Generalising the parametric transversality theorem to a foliation 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.
 A: 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?
A: The theorem says that if $F:M\times S\to N$ is transverse to $R$ then for almost every $s\in S$ the map $\phi_s:M\to N$ given by $m\mapsto F(m,s)$ is transverse to $R$. ($S$ being connected is irrelevant.) It can be proved by observing that $F^{-1}(R)$ is a submanifold of $M\times S$ and that the regular points for the projection $F^{-1}(R)\to S$ are precisely the points $(m,s)\in F^{-1}(R)$ such that $\phi_s$ is transverse to $R$ at $m$, and using Sard's Theorem.
Maybe your question is, if $F$ is transverse to some foliation of $N$ then does it follow that for almost all $s$ the map $\phi_s$ is transverse to the foliation? But this is not true. For example, take a foliation of $S$. The projection $M\times S\to S$ is a submersion, and therefore transverse to every immersed submanifold of $S$, in particular to every leaf of every foliation. But the corresponding map $\phi_s:M\to S$ is the constant map $s$, and is not transverse to the leaf containing the point $s$ except in the extreme case where leaves have codimension zero.
