The authors introduce the families of generalized quasipseudodistances in quasigauge spaces and define three kinds of dissipative set-valued dynamic systems with these families of generalized quasi-pseudodistances and with some families of not necessarily lower semicontinuous entropies. Assuming that quasigauge spaces are left
sequentially complete (but not necessarily Hausdorff), they prove that for each starting point each dynamic process or generalized sequence of iterations of these dissipative set-valued dynamic systems left converges. They also show that if an iterate of these dissipative set-valued dynamic systems is left quasiclosed, then these limit points are periodic points. Examples illustrating ideas, methods, definitions, and results are constructed.