In this paper, we consider a wireless-powered communication network (WPCN), where an energy-limited multi-antenna information source, powered by a dedicated power beacon, communicates with a mobile user (MU). The MU is equipped with a single-antenna and its mobility is characterized by the well-known random waypoint mobility model. To gain the advantages with the use of multiple antennas at the source, we adopt two popular multiple-antenna transmission schemes, namely maximal-ratio transmission (MRT) and transmit antenna selection (TAS). Differently from previous works which considered only static scenarios, this paper aims to investigate wireless power and information transfer in the scenario with a random mobile user under Nakagami-$m$ fading. It is noteworthy that a special case of our analysis, i.e., the Rayleigh fading case, has not been examined in the literature as well, which enhances the contribution value of the proposed analysis. Considering both MRT and TAS schemes, closed-form expressions for the outage probability, average delay-limit throughput, average delay-tolerant throughput, average bit error rate (BER), and throughput under average BER constraint are derived. The analysis quantifies the impact of the mobility and propagation environments, which are characterized by the path-loss exponent and multipath parameter for the PB-AP link and AP-MU link, on the performance of a WPCN. The analytical results are compared with Monte-Carlo simulations in order to validate the analysis and provide useful insights on the impact of different parameters on the system performance.