Understanding multibody interactions between colloidal particles out of equilibrium has a profound impact on dynamical processes such as colloidal self-assembly. A mechanism to dynamically tune the interactions during colloidal contacts may provide new ways to modulate suspension rheology and phase behavior. In this work, we engineer colloids coated with end-grafted polymers whose grafting sites are free to diffuse laterally along the surfaces. Polymer surface mobility enables dynamic geometric constraints between the colloids that control their packing into lamellar, string, and vesicle phases. Using optical tweezer experiments, we demonstrate precise control of dynamic interactions over a range of pico-Newton forces and seconds timescales. In a separate but related measurement, we use micro-hydrodynamic theory and optical tweezers experiments to study the transmission of fluid-mediated forces between colloids embedded within a viscoelastic medium. Memory effects from surrounding macromolecular rearrangements generate delayed transmission of hydrodynamic forces. We anticipate that our results will inform new constitutive models and advanced simulation tools to design heterogeneous fluids.