Let $M$ be a compact manifold, $F$ be a smooth manifold, and consider a homomorphism $h\colon \pi_1(M) \to {\rm Diff}(F)$. If $\widetilde{M}$ stands for the universal covering of $M$, we have a diagonal action of $\pi_1(M)$ on the product $\widetilde{M}\times F$, and the foliation $\{\widetilde{M}\times \{y\}\}_{y\in F}$ induces a foliation on the smooth quotient $M(h)=(\widetilde{M}\times F)/\pi_1(M)$.
It seems that
the image of $\widetilde{M}\times \{y\}$ in $M(h)$ is a compact leaf of the induced foliation if and only if $y$ is fixed by all values of $h$.
See for example this answer. I have tried looking for a proof of this in about a dozen books on foliation theory. It seems like a folklore result. Since no one could be bothered to write down a proof of this, I assume it must be trivial, but I would still like a reference for a complete proof.
When $\dim F=1$, it also seems like the above is a particular case of (or at least related to)
if $L$ is a compact leaf of a codimension-one foliation on $M$, then compact leaves sufficiently close to $L$ correspond to fixed points of the leaf holonomy of $L$.
It would be convenient to have a reference for this last claim too. Thanks.