(Sorry for what is probably a rather foundational PL-topology question.) By a triangulation of a manifold $M$, I mean a homeomorphism with the geometric realization of a simplicial complex, $h: |K| \to M$. I am wondering if I am given two triangulations $h_0 : |K_0| \to M \times \{0\}$ and $h_1 : |K_1| \to M \times \{1\}$, can I find a triangulation $h : |K| \to M \times I$ of $M \times [0,1]$ that extends $h_0$ and $h_1$?

By this I mean that $K_0$ and $K_1$ both include into $K$ and composing the geometric realization of either of these inclusions with $h$ commutes with $h_0$ and $h_1$.

I am happy to assume $M$ is compact, orientable, and smooth. I don't particularly care about the triangulations being PL (i.e. I don't care if the links of vertices are spheres).