I was thinking about another problem, and at one point it would have been convenient to be able to say this:
Let $A$ be closed in the metric space $X$, and $F: A \times I \rightarrow X$ a deformation of $A$ to the point $a$. There is a deformation $F': A \times I \rightarrow X$ deforming $A$ to $a$ such that if $s < t$ and $f_s(x) = f_s(y)$ then $f_t(x) = f_t(y)$.
In other words, if the homotopy identifies two points $x, y \in X$, then those points stay identified for the rest of the homotopy. Here it's assumed that $A$ is contractible and $F$ is a homotopy equivalence. For general homotopies between closed subsets it's false; for example, the reverse homotopy of a contraction from a non-degenerate set. I don't know if "closed" is required; the same question could be asked for open sets, or for spaces more general than metric.
Another type of counterexample for general subsets would be identifying the center point of two closed discs, and homotoping one boundary circle to the other. It's actually a bit more complicated to prove than you might expect, but you can't get an "inclusion-strict" homotopy equivalence.
But counterexamples seem kind of hard to come up with, generally. Has this concept been studied before?