If $X$ is a $C$-continuous semi-martingale then $$ \int_0^{C_t} H_s\, dX_s = \int_0^t H_{C_s}\,dX_{C_s}. $$ As far as I'm aware, this and its consequences are the only relation between stochastic integrals and time-changes.
If $f$ is increasing, deterministic and sufficiently smooth let $Y^f_t = X_{f(t)}$. Is there a relationship between $dY^f$ at time $t$ and $dX$ at time $f(t)$?