A piecewise linear tristable reaction-diffusion equation under external forcing of periodic type is considered. A special feature of the forcing is that the force moves together with the traveling wave. Front velocity equations are obtained analytically using matching procedures for the front solutions. It is noted that there is a restriction in building of null-cline. For each choice of outer branches of null-cline the middle interfacial zone should not exceed some critical value. When this zone is larger the front does not exist. It is found that in the presence of forcing there exists a set of front solutions with different phases (matching point coordinates). The periodic forcing produces a change in the velocity-versus-phase diagram. For a specific choice of wave number, there is a bubble formation which corresponds to additional solutions when the velocity bifurcates to form three fronts.