The direct analogue of Kirby diagrams exist of course but they tend to be things people have difficulty visualizing. For example, attaching a 3-handle to a 5-ball amounts to embedding an $S^2$ in $S^4$. But embeddings of $S^2$ in $S^4$ are things that don't have easy diagramatic representations. There is Kamada's "Braids and knot theory in dimension 4", Carter and Saito's "Knotted surfaces" book and a few other techniques. An example that gets at the heart of the problem is that there are known knots in $S^4$ called Cappell-Shaneson knots but as of yet, nobody has found an explicit way of visualizing them in $S^4$.
In high dimensions people tend to prefer describing their manifolds via their invariants, rather than in terms of explicit constructions. This is because of the s-cobordism theorem, which in a sense tells you that that's all you really need.
Conversely, the usage of handle / surgery descriptions in dimension 4 is largely due to the lack of an s-cobordism theorem in that range.