gt.geometric topology - Name for the motion of an immersion?

I have an immersion of a 2-simplicial complex S in $mathbb{R}^3$, and then a piecewise linear motion of that immersion over an interval of time [0,1].

Is there an existing name for the map $f:Stimes[0,1]tomathbb{R}^3times[0,1]$?

Update: here is a more detailed definition of f. Let $i_t$ be the immersion of S in $mathbb{R}^3$ parameterized over the piecewise linear motion: $i:Stimes[0,1]tomathbb{R}^3$. Now, extrude $mathbb{R}^3$ into a space-time $mathbb{R}^3times[0,1]$. Then the map above that I'm interested in is defined as $f(x,t)=(i_t(x),t)$.

Calling the map a homotopy seems incorrect because then the codomain should really be $mathbb{R}^3$. I'm interested in looking at the critical phenomena in a Morse theory or singularity theory sense, though I'm relatively ignorant of those fields. Perhaps there is some standard terminology to use from there.

Another possibility which came to mind was thinking of this swept immersion as a cobordism, although that didn't seem quite right since I care about the temporal ordering and resulting causality between the critical phenomena. (e.g. collisions)