  We extend the It\^o-Wentzell formula for the evolution along a continuous
semimartingale of a time-dependent stochastic field driven by a continuous
semimartingale to tensor field-valued stochastic processes on manifolds. More
concretely, we investigate how the pull-back (respectively, the push-forward)
by a stochastic flow of diffeomorphisms of a time-dependent stochastic tensor
field driven by a continuous semimartingale evolves with time, deriving it
under suitable regularity conditions. We call this result the
Kunita-It\^o-Wentzell (KIW) formula for the advection of tensor-valued
stochastic processes. Equations of this nature bear significance in stochastic
fluid dynamics and well-posedness by noise problems, facilitating the
development of certain geometric extensions within existing theories.
