  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
  \begin{gather*}
  \frac{\partial^\beta}{\partial t^\beta}u(t,x)
  = -(-\Delta_x)^\gamma u(t,x), \quad \beta,\gamma\in(0,1).
  \end{gather*}
