  We present a singular perturbation theory applicable to systems with hybrid
boundary layer systems and hybrid reduced systems {with} jumps from the
boundary layer manifold. First, we prove practical attractivity of an adequate
attractor set for small enough tuning parameters and sufficiently long time
between almost all jumps. Second, under mild conditions on the jump mapping, we
prove semi-global practical asymptotic stability of a restricted attractor set.
Finally, for certain classes of dynamics, we prove semi-global practical
asymptotic stability of the restricted attractor set for small enough tuning
parameters and sufficiently long period between almost all jumps of the slow
states only.
