For g>=4 and g'<=3*2^(g-3), every non-constant holomorphic map between moduli spaces is a forgetful map.
Ballmann, Lectures on K¨ ahler manifolds, ESI Lect
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Holomorphic maps between moduli spaces II
For g>=4 and g'<=3*2^(g-3), every non-constant holomorphic map between moduli spaces is a forgetful map.