As shared mobility solutions are developing, the next generation of e-ride-hailing systems are announced to be based on self-driving vehicles. Autonomous Mobility on Demand (AMoD) could enhance the level of service experienced by travelers thanks to lower repositioning costs and centralized control of compliant vehicles. The dispatching strategy determines system performance from AMoD and travelers' points of view. Rule-based heuristics do not exploit the full potential of the fleet because they are myopic to future requests. The rolling horizon approach allows to anticipate and make better dispatch decisions. In this study, we question the benefits for AMoD and travelers to include a short/mid-term demand prediction in an assignment-based heuristic. We define four variants of the assignment-based approach, with order matching only or joint order matching + fleet management, with impatient or productivist AVs. In a theoretical agglomeration, we test the sensitivity of horizon length on AMoD's profit, order response rate and, travelers' waiting times under the four strategies. We carefully study the outcomes of the profit-oriented heuristic in terms of equity.