In a companion paper, the theory for the evaluation of the acoustic field of a moving source in range‐independent environments was developed [P. H. Lim and J. M. Ozard, J. Acoust. Soc. Am.94, 131–137 (1993)]. In the present paper, the problem of extending the calculations to sources moving in weakly range‐dependent environments is examined. The calculations in both papers are valid for sources whose velocity is small and horizontal but otherwise arbitrary. The formalism is initially developed in the context of adiabatic mode theory without mode coupling. The corresponding equations for one‐dimensional range dependence are further developed and the paper concentrates on solutions in this case. First, the acoustic field is obtained for a point source moving in an isospeed wedge‐shaped ocean. This solution reduces exactly to previously found fields for the special cases of a uniform waveguide and of a stationary source in the wedge ocean. In the case that both source and receiver are distant from the shore, a new solution exhibiting reflections off the sloping ocean floor is presented. The acoustic field is valid for arbitrary but small horizontal source motions and the presence of reflected waves may give rise to new and interesting features in matched‐field processing problems. The remainder of the paper develops the field of a point source moving with arbitrary velocity in an ocean that is a perturbation of a uniform waveguide. The ensuing solution is then a perturbation of the solution presented in the companion paper. This field is developed explicitly for a specific bathymetry. As in the companion paper, the fields are naturally expressed in terms of retarded times. For a specific type of source motion, the acoustic field can always be recast in terms of contemporary time, and the resulting field is then in a convenient form for implementation.