Interactions between the rotating and advancing pin-shaped tool (terminated at one end with a circular—cylindrical shoulder) with the clamped welding plates and the associated material and heat transport during a friction-stir welding (FSW) process are studied computationally using a fully coupled thermomechanical finite element analysis. To surmount potential numerical problems associated with extensive mesh distortions/entanglement, an arbitrary Lagrangian—Eulerian (ALE) formulation was used, which enabled adaptive remeshing (to ensure the continuing presence of a high-quality mesh) while allowing full tracking of the material-free surfaces. To demonstrate the utility of the present computational approach, the analysis is applied to the case of FSW of AA5083 (a solid—solution strengthened and strain-hardened/stabilized Al—Mg wrought alloy). To account for the competition between plastic deformation-controlled strengthening and dynamic recrystallization-induced softening phenomena during the FSW process, the original Johnson—Cook strain and strain-rate hardening and temperature-softening material strength model is modified using the available recrystallization kinetics experimental data. Lastly, the computational results obtained in the present work are compared with their experimental counterparts available in the open literature. This comparison revealed that general trends regarding spatial distribution and temporal evolutions of various material-state quantities and their dependence on the FSW process parameters are reasonably well predicted by the present computational approach.