This paper investigates the energy-efficient hybrid beamforming design for a multi-functional integrated sensing, communications, and powering (ISCAP) system. In this system, a base station (BS) with a hybrid analog-digital (HAD) architecture sends unified wireless signals to communicate with multiple information receivers (IRs), sense multiple point targets, and wirelessly charge multiple energy receivers (ERs) at the same time. To facilitate the energy-efficient design, we present a novel HAD architecture for the BS transmitter, which allows dynamic on-off control of its radio frequency (RF) chains and analog phase shifters (PSs) through a switch network. We also consider a practical and comprehensive power consumption model for the BS, by taking into account the power-dependent non-linear power amplifier (PA) efficiency, and the on-off non-transmission power consumption model of RF chains and PSs. We jointly design the hybrid beamforming and dynamic on-off control at the BS, aiming to minimize its total power consumption, while guaranteeing the performance requirements on communication rates, sensing Cramér-Rao bound (CRB), and harvested power levels. The formulation also takes into consideration the per-antenna transmit power constraint and the constant modulus constraints for the analog beamformer at the BS. The resulting optimization problem for ISCAP is highly non-convex. Please refer to the paper for a complete abstract.