We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C 0-semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.