We analyze a measure for the degree of chaos in the magnetic fields of magnetic confinement fusion reactors by calculating the lobe fluxes of turnstiles - a method developed for characterizing transport into and out of resonance zones in Hamiltonian dynamical systems. We develop an efficient algorithm based on an action principle to calculate this quantity directly from the magnetic field, including stellarator magnetic fields which are sourced by a complicated set of three-dimensional coils. In the analyzed devices, the turnstile flux on the inboard (plasma-facing) manifolds is smaller than the turnstile flux on the outboard (wall-facing) manifolds. This indicates that the relative importance of chaos increases as the plasma facing components are moved further away from the plasma. The application of the turnstile flux calculation for the design of future reactors will be discussed.