The tendency to ergodicity of transport through heterogeneous stratified formations by a flow tilted with respect to the bedding is examined in this note. The idealized model of evenly stratified formations resembles recharge areas in naturally layered sedimentary geological structures over short distances, and transport features of more complex heterogeneous structures. Two cases are considered herein: the ergodic limit and the nonergodic regime. In the former case the theory predicts a constant asymptotic value of longitudinal dispersivity controlled by the log transmissivity integral scale. In the latter case, asymptotic results of an analytic nature are derived for the limit case of large travel times. Monte Carlo simulations are performed to study the plume evolutions for a wide range of heterogeneities and of initial size of the solute body transverse to the bedding. Results are compared with the analytical solution. It is concluded that, in the realistic case of finite initial transverse size of the plumes, ergodicity is not obeyed. Ergodic conditions, in our experiments, were not achieved even for a solute body whose dimension was 400 times the log transmissivity correlation scale. In such cases, theoretical and numerical evidence suggests that in nonergodic regimes the longitudinal dispersion coefficient tends asymptotically to zero for any initial size of the solute body.