Let $M_{g,n}$ be the moduli space of $n$-pointed curves, and $M_g[m]$ the moduli space of (unpointed) curves with $m$-level structure.

Fix $m>0$. Is it true that for $n$ large enough, there is a dominant rational map $M_{g,n}\to M_g[m]$?

I am not aware of the existence of a "natural" (i.e. functorial) map $M_{g,n}\to M_g[m]$. For my purposes, any dominant rational map would work, but I understand that it is probably difficult to construct a "non-canonical" one.