Are homotopy equivalent submanifolds also cobordant?

Question

Let $M$ be a manifold, and let $A$ and $B$ be two submanifolds of $M$ which are diffeomorphic to each other. I'll say $A$ and $B$ are homotopy equivalent in $M$ if there is a $C^1$ function $f:A \times [0,1] \mapsto M$ such that $f(\cdot,0)$ is is the inclusion map for $A$ and $f(\cdot, 1)$ is the inclusion map for $B$ up to a diffeomorphism. My question is, under what circumstances does homotopy equivalence of $A$ and $B$ imply that $A$ and $B$ are cobordant in $M$ in the sense that there is a submanifold with boundary, $C$, obeying $\partial C=A\cup B$? If $A$, $B$, $C$ need only be immersed submanifolds it seems like this might be true without further restriction, since we can define $C$ as the image of $f$, but if we require $A$, $B$, $C$ to be embedded submanifolds this construction will generically produce self-intersections in $C$ and it is not clear to me under what circumstances these can be "deformed away". Also as pointed out in the comments, even in the immersed case $f$ is not necessarily an immersion, so we still need to ask if it can be deformed into one.