The p-spinning construction works as follows:
You start with $A$ an $n$-ball embedded in an $m$-ball $B$ such that $\partial A \subset \partial B$ and $A^\circ \subset B^\circ$.
The $p$-spin of (A,B) is $\partial (A \times D^{p+1})$ in $\partial (B \times D^{p+1})$.
It is then claimed that the $p$-spin is created by sweeping interior points on a $p$-sphere while fixing the boundary.
I kinda see it is as to $x \in A^\circ$ corresponds $x \times S^p$, so indeed we sweep the point $x$ around $S^p$. What I can't see is how we an say that the boundary is fixed as in an analogous way to a point $x \in \partial A$ corresponds $x \times D^{p+1}$ and so I would say we sweep points in the boundary through a disk and not fixing it.