We investigate a superdiffusive behavior found in a quasiclassical model of a square-planar superlattice subjected to a perpendicular magnetic field. It is shown that certain accelerated domains are responsible for long trapping of tracers and setting them into near-ballistic motion. The mechanism of entrapment appears to be two-staged and multifractal. Relatively short trapping occurs in the vicinity of homoclinic tangles, created by intersections of stable and unstable manifolds of a hyperbolic fixed point, connected to itself. A structure of the quasitrap reveals families of multipulse solutions, doubly asymptotic to slow manifolds. The existence of orbits of this type was proved [G. Haller and S. Wiggins, Arch. Rational Mech. Anal. 130, 25 (1995)] for integrable two-degree-of-freedom Hamiltonian systems with perturbation. We describe mixing dynamics in this region and examine characteristic escape time scales. More prolonged quasitrapping is due to sticking to resonant multilayered island chains that are found to accelerate ballistic transport. Phase-space dynamics is analyzed. We successfully employ a renewal process formalism to relate Poincare recurrences and coordinate variance asymptotics for both quasitraps and also justify the use of this formalism for the specific case of gradually increasing average velocity.