We investigate the dynamics of a system of coupled electron billiards by using a magnetic field to dramatically modify the underlying mixed phase space. At specific values of the magnetic field the sea of chaos is drained. At these fields there exist reflected or transmitted orbits associated with maxima and minima in the experimentally observed magnetoresistance. These effects are studied by comparing the classical and quantum-mechanical phase-space dynamics leading to a basic understanding of the role of chaos in the transport in an array of billiards.