I think that a compact curve in $M_g$ provides an example. In such a family, the length of the shortest geodesic is bounded. Whereas, in a family produced by ramifying over four points in $P^1$, as two ramification points collide, there is a loop whose image downstairs stays near that pair of points, looping about them several times, and thus has arbitrarily small length. There's probably a variant of this argument with Mumford-Tate groups.