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Diffusion and Drift in Volume-Preserving Maps Nearly integrable volume-preserving maps and incompressible flows have many invariant tori on which the dynamics is quasiperiodic in the angle variables. There are also chaotic orbits in regions between tori. Although invariant tori are barriers to chaotic transport for systems with a single action, they are not for systems with multiple actions. When the nearly integrable dynamics are also symplectic or Hamiltonian, the famous theorem of Nekhoroshev guarantees that this action instability can only occur on exponentially long time scales. We study the lack of Nekhoroshev stability in general volume-preserving maps. We find a reduced model for action drift in low-rank resonances and provide supporting numerical results for some two degree-of-freedom symplectic maps and their closely related volume-preserving counterparts.