We consider the advection-diffusion transport of tracers in a one-parameter family of plane periodic flows where the patterns of streamlines feature regions of confined circulation in the shape of "cat's eyes," separated by meandering jets with ballistic motion inside them. By varying the parameter, we proceed from the regular two-dimensional lattice of eddies without jets to the sinusoidally modulated shear flow without eddies. When a weak thermal noise is added, i.e., at large Péclet numbers, several intermediate time scales arise, with qualitatively and quantitatively different transport properties: depending on the parameter of the flow, the initial position of a tracer, and the aging time, motion of the tracers ranges from subdiffusive to superballistic. We report on results of extensive numerical simulations of the mean-squared displacement for different initial conditions in ordinary and aged situations. These results are compared with a theory based on a Lévy walk that describes the intermediate-time ballistic regime and gives a reasonable description of the behavior for a certain class of initial conditions. The interplay of the walk process with internal circulation dynamics in the trapped state results at intermediate time scales in nonmonotonic characteristics of aging not captured by the Lévy walk model.