Feedback from low-excitation radio galaxies (LERGs) plays a key role in the lifecycle of massive galaxies in the local Universe; their evolution, and the impact of these active galactic nuclei on early galaxy evolution, however, remain poorly understood. We use a sample of 10 481 LERGs from the first data release of the LOFAR Two-meter Sky Survey Deep Fields, covering ∼ 25 deg2, to present the first measurement of the evolution of the radio luminosity function (LF) of LERGs out to z ∼ 2.5; this shows relatively mild evolution. We split the LERGs into those hosted by quiescent and star-forming galaxies, finding a new dominant population of LERGs hosted by star-forming galaxies at high redshifts. The incidence of LERGs in quiescent galaxies shows a steep dependence on stellar-mass out to z ∼ 1.5, consistent with local Universe measurements of accretion occurring from cooling of hot gas haloes. The quiescent-LERGs dominate the LFs at z < 1, showing a strong decline in space density with redshift, tracing that of the available host galaxies, while there is an increase in the characteristic luminosity. The star-forming LERG LF increases with redshift, such that this population dominates the space densities at most radio-luminosities by z ∼ 1. The incidence of LERGs in star-forming galaxies shows a much weaker stellar-mass dependence, and increases with redshift, suggesting a different fuelling mechanism compared to their quiescent counterparts, potentially associated with the cold gas supply present in the star-forming galaxies.