Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th configuration spaces $F(M,k)$ is not necessarily homotopy equivalent to $F(N,k)$.
In https://www.encyclopediaofmath.org/index.php/Isotopy_(in_topology), the notion "isotopy" is defined (different from the "isotopy" in geometry of submanifolds):
An isotopy is a fibrewise-continuous mapping $f:M\times[0,1]\to N\times[0,1]$ such that $f$ takes the fibre $M\times t$ homeomorphically onto a subset of the fibre $N\times t$.
Question: If $M$ is isotopic to $N$, will $F(M,k)$ be homotopy equivalent to $F(N,k)$? I want to find equivalent conditions when $F(M,k)$ will be homotopy equivalent to $F(N,k)$. Is it possible?