In this study, we solved a problem of tracer transport in heterogeneous media with a sharp contrast in properties. Two kinds of media were considered. The first was a heterogeneous medium with a highly permeable domain in the shape of a plain-parallel layer or a straight cylinder. A highly permeable domain in the second medium corresponded to a percolation system. Classical diffusion was considered for both kinds of media as the basic physical mechanism for tracer transport. Due to long residence times of tracer particles in the low-permeability domain, three transport regimes (fast classical diffusion, subdiffusion, and slow classical diffusion) occurred in the first heterogeneous medium. Similarly, in the second medium the presence of sinks (dead ends of percolation clusters) led to subdiffusion at early times, changing to slow classical diffusion at large times when the system was above the percolation threshold. For both kinds of media, the change of transport regimes with time resulted in a complex multistage structure of breakthrough curves. The structure of the more distant tail segments was found to be more strongly affected by the earlier time transport regimes.