<p>The purpose of this work is twofold. In the first part, we present some recent results concerning model parabolic problems on metric graphs. The role of interconnected structural properties such as spectral theory, Fujita phenomenon, volume growth, and functional inequalities is highlighted, also within the broader framework of metric measure spaces. In particular, we address propagation or extinction of solutions to a semilinear Cauchy-Neumann problem on regular metric trees, discussing analogies and differences with the known results for analogous problems in Euclidean and hyperbolic space. The presented results were obtained when the forcing term in the parabolic equation is of Kolmogorov-Petrovski-Piskunov type. In the last part, we prove new extinction results for the analogous problem on general metric graphs when the forcing term is of Allen-Cahn type.</p>