  Chaotic transport in Hamiltonian systems is often restricted due to the
presence of partial barriers, leading to a limited flux between different
regions in phase phase. Typically, the most restrictive partial barrier in a 2D
symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D
torus. For a 4D symplectic map we establish a partial barrier based on what we
call a cantorus-NHIM, a normally hyperbolic invariant manifold (NHIM) with the
structure of a cantorus. Using a flux formula, we determine the global 4D flux
across a partial barrier based on a cantorus-NHIM by approximating it with
high-order periodic NHIMs. In addition, we introduce a local 3D flux depending
on the position along a resonance channel, which is relevant in the presence of
slow Arnold diffusion. Moreover, for a partial barrier composed of stable and
unstable manifolds of a NHIM we utilize periodic NHIMs to quantify the
corresponding flux.
