On non-compact manifolds I think one must accept that proper Morse functions carry far less topological data than their analogous brethren on compact manifolds.

A silly illustration: let $N$ be a compact topologically-fascinating manifold, and consider the projection map $pr_2: N \times R \to R$. We see that $pr_2$ is a proper Morse function having no critical points on the noncompact $N \times R$. Moreover, the condition of having no critical points is surely an open condition, hence all Morse functions $f$ in a neighborhood of $pr_2$ (say, in compact-open topology on $C^\infty$) will have no critical points. In otherwords, the "passing through critical points means adding handles"-machinery is worthless (at least for these $f$ near $pr_2$). The value in having a proper Morse function $f$ with no critical points on some noncompact $M$ is to tell us exactly that $M$ splits as $N \times R$ (i.e. the normal bundle of a fibre is trivial). But I think it can tell us no more than that: the topology of the $N$ factor remains `hidden' from $f$.

Moreover, if one does not restrict to proper Morse functions, then I think one must expect even less! For illustration, just think of a closed manifold $M$ with some Morse function $f$ and let $Z$ be the set of critical points. Then $f$ restricts to a nonproper critical-point-free Morse function on the noncompact $M \setminus Z$. But examples show that $M \setminus Z$ does not split an $R$-factor as before. Moreover, I'll be damned if I see any way to leverage $f$ to yield any topological data on $M \setminus Z$.

Finally, one cannot ignore that our gradient flows (which, as we all learned from Milnor, are singlehandedly responsible for our much-valued deformation retracts) may not be globally defined in the non-compact case. This is probably the most important point.

In brief, I think Morse functions (proper or not) on non-compact manifolds are almost topologically useless. It should not be taken for granted that (i) a given Morse function has critical points, and (ii) that the given Morse function has enough critical points to sufficiently illuminate the topological structure of our non-compact manifold.