  We establish It\^o's formula along flows of probability measures associated
with general semimartingales; this generalizes existing results for flows of
measures on It\^o processes. Our approach is to first establish It\^o's formula
for cylindrical functions and then extend it to the general case via function
approximation and localization techniques.
  This general form of It\^o's formula enables the derivation of dynamic
programming equations and verification theorems for McKean--Vlasov controls
with jump diffusions and for McKean--Vlasov mixed regular-singular control
problems. It also allows for generalizing the classical relationship between
the maximum principle and the dynamic programming principle to the
McKean--Vlasov singular control setting, where the adjoint process is expressed
in terms of the derivative of the value function with respect to the
probability measures.
