  Motivated by recent development of mean-field systems with common noise, this
paper establishes Ito's formula for flows of conditional probability measures
under a common filtration associated with general semimartingales. This
generalizes existing works on flows of conditional measures on Ito processes
and flows of deterministic measure on general semimartingales. The key
technical components involve constructing conditional independent copies and
establishing the equivalence between stochastic integrals with respect to the
conditional law of semimartingales and the conditional expectation of
stochastic integrals with respect to copies of semimartingales. Ito's formula
is then established for cylindrical functions through conditional independent
copies, and extended to the general case through function approximations.
