I will state a very specific case: genus 5. Though it's particular, it admits a generalization to $M_g$, and I think reflects the nature of a general stratification of $M_g$.

It is known that if you have a genus five curve $C$ we've got the following disjoint familes:

-$C$ is hyperelliptic; degree 2 map $\phi:C\rightarrow\mathbb{P}^1$.

-$C$ has a degree 3 map to $\phi:C\rightarrow\mathbb{P}^1$ (equivalently, it is a plane quintic with a node).

-$C$ has a degree 4 map to $\phi:C\rightarrow\mathbb{P}^1$.

Now, how are these facts related to the degree of the equation whose zero locus is $C$?.

Let me see If I'm reading off accurately the meaning of the situation above. We're saying implicitly that there exists a space where the degree of the equation defining $C$ is 2. There exists another space where the a degree 4 equation defines $C$, so on a so forth. does it sound all right?.