Let $f\colon X \to Y$ be a uniformly continuous function. Then I think it is "well-known" that it may be approximated by a Lipschitz function, and how well one can do this depends on the modulus of continuity of $f$.
So if we have a family of uniformly equicontinuous functions, we may approximate them equally well by Lipschitz functions with a common Lipschitz constant.
I want to do this now with homotopies:
Under which conditions on $X$ and $Y$ do we have the following: given any family of uniformly equicontinuous functions $f_\alpha\colon X \to Y$, we can find an equicontinuous family of homotopies $H_\alpha\colon X \times I \to Y$ from the family $f_\alpha$ to a family of Lipschitz functions $F_\alpha$ with a common Lipschitz constant for the $F_\alpha$'s?
Note that the important point is that the homotopies may be chosen to be equicontinuous. It then follows that the distance of $F_\alpha$ to $f_\alpha$ can be bounded from above independently from $\alpha$. I'm actually not much interested in making this distance arbitrarily small, though I assume that any proof of the above statement will accomplish this too.
I would guess that if $X$ and $Y$ are locally nice enough in a uniform manner, e.g., simplicial complexes, then the statement is true. But I'm lacking a proof of this.