A simple stochastic model of solute drag by moving grain boundaries (GBs) is presented. Using a small number of parameters, the model describes solute interactions with GBs and captures nonlinear GB dynamics, solute saturation in the segregation atmosphere, and the breakaway from the atmosphere. The model is solved by kinetic Monte Carlo (KMC) simulations with time-dependent transition barriers. The non-Markovian nature of the KMC process is discussed. In this paper (which is Paper I of this work), the model is applied to planar GBs driven by an external force. The model reproduces all basic features of the solute drag effect, including the maximum of the drag force at a critical GB velocity. The force-velocity functions obtained depart from the scaling predicted by the classical models by Cahn and Lücke-Stüwe, which are based on more restrictive assumptions. The paper sets the stage for an accompanying paper [Paper II, Phys. Rev. Mater. 7, 063404 (2023)] in which the GB will be treated as a two-dimensional solid-on-solid interface.