We study the escape dynamics of test particles in general-relativistic gravitational fields generated by core-shell models, which are used in astrophysics as idealized models to observed mass distributions, such as the interior of galaxies. As a general-relativistic core-halo system, we use exact axisymmetric static solutions of Einstein's field equations which represent the superposition of a central Schwarzschild black hole (the core) and multipolar fields from external masses (the halo). We are particularly interested in the occurrence of chaos in the escape, which is characterized by a great sensitivity of the choice of escape by a test particle to initial conditions. The motion of both material particles and zero rest mass particles is considered. Chaos is quantified by the fractal dimension of the boundary between the basins of the different escapes. We find chaos in the motion of both material particles and null. geodesics, but its intensity depends strongly on the halo. We have found for all the cases we have considered that massless particles are less chaotic than massive particles.616A65066516