  In distribution theory the pullback of a general distribution by a
$C^{\infty}$-function is well-defined whenever the normal bundle of the
$C^{\infty}$-function does not intersect the wavefront set of the distribution.
However, the Colombeau theory of generalized functions allows for a pullback by
an arbitrary c-bounded generalized function. It has been shown in previous work
that in the case of multiplication of Colombeau functions (which is a special
case of a $C^{\infty}$ pullback), the generalized wave front set of the product
satisfies the same inclusion relation as in the distributional case, if the
factors have their wavefront sets in favorable position. We prove a microlocal
inclusion relation for the generalized pullback (by a c-bounded generalized
map) of Colombeau functions. The proof of this result relies on a stationary
phase theorem for generalized phase functions, which is given in the Appendix.
Furthermore we study an example (due to Hurd and Sattinger), where the pullback
function stems from the generalized characteristic flow of a partial
differential equation.
