<p>Parity–time symmetry has revolutionized wave and energy transport control in non-Hermitian systems, yet has so far been mostly explored in static phases, where a system’s behaviour is locked into a fixed-symmetric or broken-symmetry phase. The vast potential of time-domain dynamics has remained largely untapped. Here we introduce the concept of temporal anti-parity–time symmetry, a principle that allows the transport dynamics of a system to be actively shaped in real time. Rather than designing static phases, we influence the timing of non-Hermitian phase transitions, making the system’s temporal evolution itself a programmable degree of freedom. Through the dynamic control of material properties and convective flow, we dictate the exact moments these transitions occur, thereby controlling the entire transport history of the system. This temporal control achieves highly tunable field localization and realizes counterintuitive thermal transport, enabling temperature profiles to move forwards with convection, backwards against it or remain trapped at arbitrary locations. Our findings extend non-Hermitian physics into the time domain and establish a framework for on-demand wave and energy transport.</p>