Plastic deformation in elemental BCC metals and dilute alloys is controlled by the slower of the kink pair nucleation and kink migration processes along screw dislocations. In alloys nucleation is facilitated and migration inhibited, leading to a concentration- and temperature-dependent transition from nucleation dominance to migration dominance. Here, an analytical statistical model for the stress-dependent kink migration barrier in dilute BCC alloys is developed and validated. The barrier depends only on a clearly-defined solute/screw dislocation interaction parameter, the kink width, and dislocation length between jogs. The analytic model is extensively validated via fully atomistic nudged-elastic band calculations and stochastic simulations in a model Fe-Si alloy. Combined with a recent validated double-kink nucleation theory, a fully-analytic model for the temperature- and concentration-dependent flow stress is obtained that includes the transition from nucleation to migration control. The overall model is applied to Fe-Si and W-Re using independently-determined material properties and good agreement is obtained with experiments over a range of concentrations and temperatures. Overall, the two theories represent a unified, fully-statistical, parameter-free understanding of screw dislocation strength in dilute BCC alloys.