  We develop semiclassical methods for studying bubble nucleation in models
with parameters that vary slowly in time. Introducing a more general rotation
of the time contour allows access to a larger set of final states, and
typically a non-Euclidean rotation is necessary in order to find the most
relevant tunneling solution. We work primarily with effective quantum
mechanical models parametrizing tunneling along restricted trajectories in
field theories, which are sufficient, for example, to study thin wall bubble
nucleation. We also give one example of an exact instanton solution in a
particular Kaluza-Klein cosmology where the circumference of the internal
circle is changing in time.
