Some experimental results in the study of disordered systems, polymeric fluids, solutions of micelles and surfactants, ionic-glass conductors, and others show a hydrodynamic behavior labeled "anomalous" with properties described by some kind of fractional power laws in place of the standard ones. This is a consequence of the fractal-like structure that is present in these systems of which we do not have a detailed description, thus impairing the application of the conventional ensemble formalism of statistical mechanics. In order to obtain a physical picture of the phenomenon for making predictions which may help with technological and industrial decisions, one may resort to different styles (so-called nonconventional) in statistical mechanics. In that way can be introduced a theory for handling such impaired situations, a nonconventional mesoscopic hydrothermodynamics (MHT). We illustrate the question presenting an application in a contracted description of such nonconventional MHT, consisting in the use of the Renyi approach to derive a set of coupled nonstandard evolution equations, one for the density, a nonconventional Maxwell-Cattaneo equation, which in a limiting case goes over a non-Fickian diffusion equation, and other for the velocity in fluids under forced flow. For illustration the theory is applied to the study of the hydrodynamic motion in several soft-matter systems under several conditions such as streaming flow appearing in electrophoretic techniques and flow generated by harmonic forces arising in optical traps. The equivalence with Lévy processes is discussed and comparison with experiment is done.