In this work, we address the problem of enhancing Quality-of-Service (QoS) in power constrained, mobile AF relay beamforming networks, by optimally exploiting relay mobility. We consider a time slotted system, where the relays update their positions before the beginning of each time slot. Adopting a spatiotemporal stochastic field model of the wireless channel, we propose a novel 2-stage stochastic programming formulation for specifying the relay positions at each time slot, such that the QoS of the network is maximized on average, based on causal CSI and under a total relay transmit power budget. Via the Method of Statistical Differentials, the motion control problem considered is shown to be approximately equivalent to a set of simple subproblems, which can be solved in a distributed fashion, one at each relay. It is also shown that, under mild, natural assumptions on the stochastic nature of the channel model considered, the proposed 2-stage framework exhibits a striking property: The average network QoS increases across times slots. Numerical simulations are presented, corroborating the efficacy of the proposed approach and confirming its properties.