In Hirsch's differential topology, we find the following theorems on page 31:
4.1 Theorem: Let $M$ be a $C^r$ $\partial$-manifold and $N$ a $C^r$ manifold, $r \geq 1$. Let $f : M \to N$ be a $C^r$ map. If $y \in N - \partial N$ a regular value for both $f$ and $f|_{\partial M}$, then $f^{-1}(y)$ is a neat $C^r$ submanifold of $M$.
4.2 Theorem: Let $A \subset N$ be a $C^r$ submanifold and $f : M \to N$ a $C^r$ map. Suppose $\partial A = \emptyset$ and $f$, $f|_{\partial M}$ are both transverse to $A$. Then $f^{-1}(A)$ is a $C^r$ submanifold with boundary $f^{-1}(A) \cap \partial M$.
The proofs of these results are left as exercise. I know how to prove them for boundaryless manifolds using the constant rank theorem, but I'm not sure at all how to adapt this for boundary manifolds. Any hints or references would be greatly appreciated.