  The presence of spatial inhomogeneity in a nonlinear medium restricts the
formation of Solitary Waves (SW) on a discrete set of positions whereas a
nonlocal nonlinearity tends to smooth the medium response by averaging over
neighboring points. The interplay of these antagonistic effects is studied in
terms of SW formation and propagation. Formation dynamics is analyzed under a
phase space approach and analytical conditions for the existence of either
discrete families of Bright SW or continuous families of Kink SW are obtained
in terms of Melnikov's method. Propagation dynamics are studied numerically and
cases of stable and oscillatory propagation as well as dynamical transformation
between different types of SW are shown. The existence of different types and
families of SW in the same configuration, under appropriate relations between
their spatial width and power with the inhomogeneity and the nonlocality
parameters, suggests an advanced functionality of such structures that is quite
promising for applications.
