I learned about the push-pull construction from a video lecture by Freedman in which it is explained starting around 39:08. It is somewhat long and technical to describe in detail, but the main idea is that it is a technique to control an homeomorphism in one linear direction.
The example application is to show that if $X$ and $Y$ are compact metric spaces such that $X\times\Bbb R$ is homeomorphic to $Y\times\Bbb R$, then $X\times S^1$ and $Y\times S^1$ are also homeomorphic. The way this is done is by taking an homeomorphism $h\colon X\times\Bbb R\to Y\times\Bbb R$ and using the push-pull construction to modify it to another homeomorphism between the two spaces which is periodic in the $\Bbb R$ coordinate, so that it descends to an homeomorphism $X\times S^1\to Y\times S^1$.
Morse used the construction in 1960 to establish the Schoenflies conjecture "Every bicollared embedding $S^d\to\Bbb R^{d+1}$ extends to an embedding $D^{d+1}\to\Bbb R^{d+1}$". In particular he used the push-pull construction to show that every such bicollared embedding admits a so called flat spot after composing with a self-homeomorphism of $\Bbb R^{d+1}$, and the conjecture had already been established by Mazur for bicollared embeddings with a flat spot.
However Freedman mentions that the push-pull construction was not introduced by Morse, and in fact that it was already well known by the time he used to to settle Schoenflies's conjecture, hence my question: who invented the push-pull construction?
