  Moduli spaces of global $\mathbb G$-shtukas play a crucial role in the
Langlands program for function fields. We analyze their functoriality
properties following a change of the curve and a change of the group scheme
$\mathbb G$ under various aspects. In particular, we prove two finiteness
results which are of interest in the study of stratifications of these moduli
spaces and which potentially allow the formulation of an analog of the
Andr\'{e}-Oort conjecture for global $\mathbb G$-shtukas.
