  The paper is concerned with the change of probability measures $\mu$ along
non-random probability measure valued trajectories $\nu_t$, $t\in [-1,1]$.
Typically solutions to non-linear PDEs, modeling spatial development as time
progresses, generate such trajectories. Depending on in which direction the map
$\nu\equiv\nu_0\mapsto\nu_t$ does not exit the state space, for $t\in [-1,0]$
or for $t\in [0,1]$, quasi-invariance of the measure $\mu$ under the map
$\nu\mapsto\nu_t$ is established and the Radon-Nikodym derivative of
$\mu\circ\nu_t$ with respect to $\mu$ is determined. It is also investigated
how Fr\'echet differentiability of the solution map of the PDE can contribute
to the existence of this Radon-Nikodym derivative. The first application is a
certain Boltzmann type equation. Here the Fr\'echet derivative of the solution
map is calculated explicitly and quasi-invariance is established. The second
application is a PDE related to the asymptotic behavior of a Fleming-Viot type
particle system. Here quasi-invariance is obtained and it is demonstrated how
this result can be used in order to derive a corresponding integration by parts
formula.
