In this chapter, we systematically explore various time asymptotic regimes (asymptotic stability, sweeping, scattering theory) for the Kato-Voigt semigroups constructed in Chap. 2. Such semigroups originate from the classical 1954 Kato’s work on Kolmogorov’s differential equations and were subsequently revisited by several authors. This abstract class of semigroups encompasses numerous concrete applications in Transport Theory, Mathematical Biology, and related fields.