Let $E$ be an elliptic curve over a number field $K$. The difference between the semistable Faltings height $h_F(E)$ of $E$ and the height $h(j_E)$ of the $j$-invariant of $E$ can be bounded in terms of $h(j_E)$, see for example Proposition 2.1 in Silverman ``Heights and Elliptic Curves''.

Are similar results known when we consider a more general abelian variety instead of an elliptic curve, and the height of the corresponding point on some moduli space in place of the height of the $j$-invariant? I would be particularly interested in the case of families of abelian varieties (with appropriate extra structure) parametrised by some fixed Shimura curve.