I imagine this is pretty much standard, but surely someone here will be able to provide useful references...

Suppose $X$ is a topological space. Let me say that two triangulations $T$ and $T'$ of $X$, are homotopic if there is a triangulation on $X\times [0,1]$ which induces $T$ and $T'$ on $X\times\{0\}$ and on $X\times\{1\}$, respectively.

• This rather natural equivalence relation has probably been studied. Does it have a standard name? Are there standard references?

• In particular, can this equivalence be redefined by using local moves, ie, as the transitive symmetric closure of a (small) set of local changes in triangulations?