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The design of polycrystalline alloys hinges on a predictive understanding of the interaction between the diffusing solutes and the motion of the constituent crystalline interfaces. Existing frameworks ignore the dynamic multiplicity of and transitions between the interfacial structures and phases. Here, we develop a computationally-accessible theoretical framework based on short-time equilibrium fluctuations to extract the drag force exerted by the segregating solute cloud. Using three distinct classes of computational techniques, we show that the random walk of a solute-loaded interface is necessarily non-classical at short time-scales as it occurs within a confining solute cloud. The much slower stochastic evolution of the cloud allows us to approximate the short-time behavior as an exponentially sub-diffusive Brownian motion in an external trapping potential with a stiffness set by the average drag force. At longer time-scales, the interfacial and bulk forces lead to a gradual recovery of classical random walk of the interface with a diffusivity set by the extrinsic mobility. The short-time response is accessible via {\it ab-initio} computations, offering a firm foundation for high throughput, rational design of alloys for controlling microstructural evolution in polycrystals, and in particular for nanocrystalline alloys-by-design.