Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\dots 4\}$.

It is known ("On attaching 3-handles to a 1-connected manifold" by Bruce Trace) that if $M$ is simply connected and $\partial_+ M$ is empty or connected, $M_4$ is essentially determined by $M_2$. Or put differently, $M$ is completely determined by $\partial_- M$, $\partial_+ M$ and $M_2$.

What can we say for general $M$? Does the attaching map of a 3-handle contain nontrivial data? Are there Kirby diagrams with 3-handle attachments? (I've never seen 3-handles represented in Kirby diagrams, probably because in many cases, only their number matters.)