Let $M_g^{ct}$ denote the moduli stack of compact type genus $g$ stable curves and $A_g$ the moduli stack of principally polarized $g$-dimensional abelian varieties.

Can someone provide a reference for the fact that the Torelli map extends to a map $M_g^{ct} \to A_g$ which is proper?

Here are some comments: This is stated, for example, in Section 1.3 of "The Torelli locus and special subvarieties" by Ben Moonen and Frans Oort. I believe I have found a reference, but am not completely certain if I'm interpreting things correctly. In "Compactified Jacobians and the Torelli Map" by Alexeev http://alpha.math.uga.edu/~valery/preprints/cjtm.pdf Corollary 5.4, it is stated that there are compactifications $\overline{M}_g$ and $\overline{A}_g$ such that the Torelli map extends to a map of their coarse spaces. I believe the constructed map actually defines a map on stacks (and not just coarse spaces) which would imply the map $M_g^{ct} \to A_g$ (which is the restriction of $\overline{M}_g \to \overline{A}_g$) is also proper. So it would be enough to answer the following question affirmatively: Is it correct that the map in Corollary 5.4 is actually coming from a map of stacks and not just a map of coarse spaces?