A large amount of binary fluid mixture in the homogeneous phase near the demixing critical point can include a small spherical droplet of a third fluid-component confined by a sharp interface. Conversely, a droplet consisting of the mixture can be immersed in the third fluid-component. In either of the situations, we assume that the third component attracts one mixture component more than the other via short-range interactions. The adsorption layer, enriched with the preferred component, appears on the mixture side of the interface and can thicken significantly because of large susceptibility. The preferential adsorption affects flows, causing the drag coefficient to deviate. We use the hydrodynamics based on a coarse-grained free-energy functional to calculate the deviation, while neglecting the weak singularity of mixture viscosity. When the mixture lies inside the droplet, the ratio of the deviation changes nonmonotonically as the ratio of the ambient viscosity to the droplet viscosity increases. When the mixture lies outside, the deviation ratio increases with the viscosity ratio and can be considerably larger for a droplet than for a rigid sphere. Then, if a composition gradient is imposed, a force-free droplet undergoes diffusiophoresis, whose mobility can be considerably larger in magnitude than that of a rigid sphere. These results can be utilized in future applications in the droplet microfluidics. We also discuss probable power-law dependences of the drag coefficient and the mobility on the reduced temperature. The suggested power for the mobility is connected with the universal order-parameter profile in the surface critical behavior.
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