In this paper, a phase field dislocation dynamics model is employed to predict the energetically favorable pathways taken by initially screw-character, long dislocations as a function of temperature and driving stress in TaNbTi and MoNbTi multi-principal element alloys (MPEAs). The simulations resolve the critical stress at which glide becomes jerky to smooth, the changes in the glide mechanisms as temperature increases, and the local impact of variations in composition-dependent energetic barriers. It is shown that the jerky-to-smooth stress exhibits a two-stage response, where it decays with temperature at low temperatures and transitions to an athermal regime at high temperatures, like that measured for these alloys. The analysis elucidates the changes in the glide processes responsible for the onset of the athermal regime in critical stress and shows a close connection to experimentally measured athermal temperatures, suggesting that screw dislocations might impact the high-temperature strength of these alloys.