A long Josephson junction with a spatially varying inductance is a physical manifestation of a modified sine-Gordon equation with parametric perturbation. Soliton propagation in such Josephson junctions is discussed. First, for an adiabatic model where the inductance changes smoothly compared with soliton size, transmission or reflection of the soliton is described using a simple energy analysis. Next, the soliton propagation is solved on the basis of a perturbation theory constructed by McLaughlin and Scott. Radiation as well as soliton trajectories are presented numerically. Agreement between such solutions and the results of direct numerical integration by means of a finite-difference method is excellent.