Here's an example I learned from John Franks. This is a nice example because it's used to produce an example of Smale's sphere eversion problem.
Consider the Boy's surface. This is an immersion of $\mathbb{R}P^2$ into $\mathbb{R}^3$. If you look at its normal bundle, there's no sense of +1 or -1 (because it's non-orientable) but you can look at its associated unit sphere bundle. This is the orientation double cover, immersed into $\mathbb{R}^3$. By scaling the fibers of the normal bundle from 1 to 0, you see the "covering homotopy" of $S^2$ onto the Boy's surface ($\mathbb{R} P^2$), as you ask for.
So that's the thing you're looking for, but let's go further--instead of just scaling from 1 to 0, scale from 1 to -1. This is a homotopy, through immersions, of $S^2$ to itself, and it leads to one way in which you can evert the sphere (i.e., turn it inside out). I think this strategy originally came from Shapiro and Phillips, though any historical corrections are welcome!