Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $f_2$, respectively. I am wondering if there will always be some submanifold $N^{n-1} \subset X \times I$ with boundary equal to these two embeddings of $M$ into $X \times I$.

If we choose a homotopy between $f_1$ and $f_2$, say $H$, then we can look at the track of the homotopy $M \times I \to X \times I$, but this will not be an embedded $N$. This then shows that the two different images of $M$ are homologous in $X \times I$ - is there some sort of Steenrod realization thing that allows we to conclude that there is such a desired $N$? Maybe I need assumptions on $n$...

For what its worth, I'm really interested in the case where everything is connected and oriented.