Two-dimensional computations of dispersed multiphase flows involving complex geometries are presented. The numerical algorithm is based on the front-tracking method in which one set of governing equations is written for the whole computational domain and different phases are treated as a single fluid with variable material properties. The front-tracking methodology is combined with a newly developed finite volume solver based on dual time-stepping, diagonalized alternating direction implicit multigrid method. The method is first validated for a freely rising drop in a straight channel, and it is then used to compute a freely rising drop in various constricted channels. Interaction of two buoyancy-driven drops in a continuously constricted channel is also presented.