We look at estimates for the Green’s function of time-fractional evolution equations of the form D_0+∗^ν u = Lu, where D^ν_0+∗ is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y^−1−β for β ∈ (0, 1), and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D₀^β u = Lu in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D₀^β u = Ψ(−i∇)u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α. Thirdly, we obtain local two-sided estimates for the Green’s function of D₀^β u = Lu where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ, as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D₀^(ν,t) u = Lu, where D^(ν,t) is a Caputo-type operator with variable coefficients.