Temperature-modulated differential scanning calorimetry (TMDSC) and broadband dielectric spectroscopy (BDS) were employed to study the glass transition, size of the cooperative rearranging regions (CRRs), crystallization kinetics, and dielectric relaxation response of nanocomposites constituted by chain-extended poly(L-lactide) (PLLA) and carboxylated carbon nanotubes (f-CNTs). The CRR size and the number of relaxing structural units decreased in the presence of crystals during isothermal crystallization. All samples displayed both a primary (alpha) and secondary (beta) relaxation in BDS spectra. The relaxation dynamics of PLLA chains was barely affected by the presence of the f-CNT. Constrained polymer chains and thickness of interphase (t(i)) were measured using dielectric spectra in tan delta representation. t(i) values were found to be 46 and 24 nm for sample containing 0.2 and 0.5% weight fraction of f-CNT, respectively. All samples underwent partial crystallization (with roughly 30% of final crystalline fraction) some 15 or 20 degrees above their glass-transition temperature (T-g). Crystallization leads to a fragile-to-strong transition in the temperature dependence of the cooperative alpha relaxation and to the increased visibility of a Maxwell-Wagner-Sillars (MWS) interfacial relaxation, which appears to be present in all samples. The heterogeneity of the polymeric samples was quantified in terms of a new parameter, the heterogeneity index (H). (c) 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 222-233