We identify, through a change of variables, solution operators for evolution equations with generators given by certain simple first-order differential operators acting on Fock spaces. This class of operators properly includes, through unitary equivalence, the well-studied set of elliptic and weakly elliptic quadratic differential operators acting on \(L^2(\Bbb{R}^n)\). We demonstrate a variety of sharp results on boundedness, decay, and return to equilibrium for these solution operators, connecting the short-time behavior with the range of the symbol and the long-time behavior with the eigenvalues of these operators. This is particularly striking when it allows for the definition of solution operators which are compact and regularizing for large times for certain operators whose spectrum is the entire complex plane.