In this paper we present a model that studies domain wall propagation in geometrically modulated cylindrical nanostructures of small radii. Thus, we use the thin wire approximation, i.e., the magnetization is assumed constant on a cross section perpendicular to the longitudinal direction (z) of the nanostructure; we deal with transverse domain walls. We obtain general expressions for the average fields in these cross sections: Zeeman, anisotropy, exchange, and magnetostatic, and thus solve the Landau-Lifshitz-Gilbert equation for the dynamics of the magnetization. This effective one-dimensional model involves a dependence of the average exchange and dipolar fields on the changing geometry through local (similar to a nonadiabatic spin transfer term) and nonlocal kernels (written in terms of special functions), respectively. We do examine the effect of "generic" obstacles on the domain wall: localized changes of radius, bumps, and necks. The speed of propagation depends on these geometric obstacles and interestingly enough one may stop the propagation of the wall by adjusting the parameters of the geometry. Thus, one sees the potential of using geometry as a way to control domain wall motion.