In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related by blow-ups and blow-downs.
I wonder if such wonderful theorem has a 4-dimensional generalization. We do have basic moves for the Kirby diagrams from handle slidings and cancellations, but do they form a complete set of moves that characterize smooth 4-manifolds?
If such question is too hard, what can we say about for the more restricted cases (e.g. connected smooth closed oriented 4-manifold, whose Kirby diagram can be made simpler)?