The introduction of a drone-based mobile sink into wireless sensor networks (WSNs), which has flexible mobility to move to each sensor node and gather data with a single-hop transmission, makes cumbersome multi-hop transmissions unnecessary, thereby facilitating data gathering from widely-spread sensor nodes. However, each sensor node spends significant amount of energy during their idle state where they wait for the mobile sink to come close to their vicinity for data gathering. In order to solve this problem, in this paper, we apply a wake-up receiver to each sensor node, which consumes much smaller power than the main radio used for data transmissions. The main radio interface is woken up only when the wake-up receiver attached to each node detects a wake-up signal transmitted by the mobile sink. For this mobile and on-demand data gathering, this paper proposes a route control framework that decides the mobility route for a drone-based mobile sink, considering the interactions between wake-up control and physical layer (PHY) and medium access control (MAC) layer operations. We investigate the optimality and effectiveness of the route obtained by the proposed framework with computer simulations. Furthermore, we present experimental results obtained with our test-bed of a WSN employing a drone-based mobile sink and wake-up receivers. All these results give us the insight on the role of wake-up receiver in mobile and on-demand sensing data gathering and its interactions with protocol/system designs.