I do not think such semigroups have been extensively studied.
I have seen such semigroups (and, more generally, evolution families corresponding to the non-autonomous problem) in
A. Lunardi, M. Geissert, Asymptotic behavior and hypercontractivity in nonautonomous Ornstein-Uhlenbeck equations. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 85--106.