Ito's formula for flows of measures on semimartingales

We establish Ito's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on Ito processes. Our approach is to first establish Ito's formula for cylindrical functions and then extend it to the general case via function approximation and localization techniques. This general form of Ito'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.