We introduce a problem in which demands arrive stochastically on a line segment, and upon arrival, move with a fixed velocity perpendicular to the segment. We design a receding horizon service policy for a vehicle with speed greater than that of the demands, based on the translational minimum Hamiltonian path (TMHP). We consider Poisson demand arrivals, uniformly distributed along the segment. For a fixed segment width and fixed vehicle speed, the problem is governed by two parameters; the demand speed and the arrival rate. We establish a necessary condition on the arrival rate in terms of the demand speed for the existence of any stabilizing policy. We derive a sufficient condition on the arrival rate in terms of the demand speed that ensures stability of the TMHP-based policy. When the demand speed tends to the vehicle speed, every stabilizing policy must service the demands in the first-come-first-served (FCFS) order; and the TMHP-based policy becomes equivalent to the FCFS policy which minimizes the expected time before a demand is serviced. When the demand speed tends to zero, the sufficient condition on the arrival rate for stability of the TMHP-based policy is within a constant factor of the necessary condition for stability of any policy. Finally, when the arrival rate tends to zero for a fixed demand speed, the TMHP-based policy becomes equivalent to the FCFS policy which minimizes the expected time before a demand is serviced. We numerically validate our analysis and empirically characterize the region in the parameter space for which the TMHP-based policy is stable.