Actin polymerization drives the motion of cells and various pathogens, through three main interconnected mechanisms: (i) formation and assembly of the actin filaments network, (ii) its protrusive growth, and (iii) disassembly. This network arises from intricate biological processes that convert globular actin (G-actin) in the cytosol into structured networks of filaments (NF-actin). To provide insights on these mechanisms, we put forward a joint experimental and computational study. A tailored experimental campaign investigated how drug-dependent modulation of actin impacts 2D mesenchymal migration. In particular, the macroscopic effects of Cytochalasin D and Jasplakinolide on cell shape and migration were evaluated and compared to those of an untreated control. Modelling and simulations are based on a system of partial differential equations (PDEs), grounded in the theory of continuum thermo-mechanics. Our formulation integrates the transport, chemical transformation, and mechanical behaviour of actin, with particular emphasis on the mechanical properties of the filament network. We propose that the polymerization of G-actin into NF-actin induces localized mechanical swelling near nucleation sites, a phenomenon that plays a key role in generating actin-driven movement. In this framework, NF-actin formation is represented by a single chemical reaction capturing nucleation, branching, and cross-linking. The resulting PDEs constitute a thermodynamically consistent, multiphysics continuum model that extends the classical Larché-Cahn theory for chemo-transport-mechanical coupling. Such equations were recast in weak form and solved using the finite element library deal.ii, enabling high-performance computing (HPC) simulations. Numerical results were successfully compared with experimental data, showing good agreement and supporting the proposed hypothesis.