This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space ∂^β /_∂tβ u(t, x) = −(−Δ_x)^γu(t, x), β, γ ∈ (0, 1).