We combine microdielectrometry (MD) sensors with polymers showing dielectric normal-mode relaxation (type A polymers) to study multicomponent diffusion in melts of entangled, linear, flexible chains. The term "normal-mode microdielectrometry" (NMMD) reflects this particular combination of ingredients. We consider cis-polyisoprene (cis-PI), a well-characterized type-A polymer. Using reptation theory and a linear mixing rule, normal-mode dielectric loss spectra of polydisperse cis-PI samples can be inverted to yield the entire molecular weight distribution (MWD). MD sensors enable probing the dielectric response of the material residing in the immediate vicinity (0(10 mu m)) of a solid surface. Therefore, the combination of type A polymers and MD sensors allows us to follow the evolution of the MWD of the material in the immediate proximity of a solid surface. We study the diffusive dissolution of a thin layer of a polydisperse, high-MW cis-PI melt, deposited at the sensor surface, into a polydisperse, semiinfinite medium of smaller chains. The data are first analyzed using a Fickian diffusion model with a molecular weight dependent effective diffusivity. The results are consistent with swelling of the high-MW melt caused by diffusive penetration of smaller chains. We also propose an extension of the Kramer-Sillescu theory of diffusion to polydisperse systems and contrast the results to the kinetic-theory equations of Bird et al. An analytical solution for the evolving near-surface MWD is obtained for a simplistic version of the Bird et al. model. Although showing qualitatively correct trends, the model is unable to quantitatively describe the experimental data.