The commercially available partially bio-based poly(butylene succinate) (PBS) and poly(butylene succinate-co-adipate) (PBSA) are subjected to prolonged and multiple extrusion cycles to investigate their thermal behavior. Empirically, both PBS and PBSA form branches at 190°C, with PBS possessing a higher tendency for branching than PBSA as studied by rheology and size exclusion chromatography (SEC). In each case, the branching is favored by fumaric acid moieties, making both PBS and PBSA susceptible to accelerated thermal oxidation. Indeed, the NMR signal attributed to fumaric acid disappears upon thermal processing of PBS and PBSA. Presumably, the bio-based succinic acid used contains minor quantities of fumaric acid, yet still sufficient to cause this surprising processing behavior. The branching of both polymers is suppressed by stabilizers such as phenolic antioxidants, as proven by rheology and SEC. This is complemented by nuclear magnetic resonance (NMR), revealing that the fumaric acid signal is still well-resolved in the stabilized processed samples.