  We provide a general It\=o\,-Wentzell formula for a random field of maps on
the Wasserstein space of probability measures, defined by continuous
semimartingales, and evaluated along the flow of conditional distributions of
another continuous semimartingale. Our method follows standard arguments of
It\=o calculus, and thus bypasses the approximation by empirical measures
commonly used in the existing literature. As an application, we derive the
dynamic programming equation for a mean field stochastic control problem with
common noise.
