Generalizations of the handle trading techniques

Yes; see Smale, On the structure of manifolds (Amer. J. Math. 84 1962 387–399) where it's shown that in high dimensions, you can eliminate handles under various connectivity assumptions. The h-cobordism theorem is a special case. For index greater than 1, one usually doesn't do handle-trading, because you can in fact just do handle cancellation, which is simpler. The reason for doing handle trading for 1-handles (and dually, for (n-1)-handles) instead of cancellations is to avoid tricky issues related to presentations of the fundamental group, related to the Andrews-Curtis conjecture.

You might find the paper by C.T.C Wall: Geometrical connectivity I, J. London Math. Soc. 3 (1971), p. 597-604, interesting. What Wall proves, entirely by handle trading, is that if $W:M_0 \to M_1$ is an $n$-dimensional cobordism and the inclusion $M_0\to W$ is r-connected, then you can built W from M_0 using only handles of index $\geq r+1$, provided that $r \leq n-4$.