Two series of newly synthesized sustainable block copolymers based on poly(butylene succinate) (PBSu) and polylactide (PLA) were studied. The copolymers were synthesized by a ring-opening polymerization of PLA in the presence of two initial PBSu of low molar mass. We focused on the effects of the PBSu/PLA ratio (1/99 up to 15/85), chain length and initial PBSu length on the final thermal transitions in the copolymers with an emphasis on molecular mobility/dynamics and subsequently on crystallization. Both aspects are considered relevant to the final materials performance, as well as facilitation of polymer renewability. Calorimetry and dielectric spectroscopy were the main investigation tools. In the amorphous state (i.e., in which the direct effects of copolymer structure are assessable), the segmental mobility of neat PLA was significantly faster in the copolymers. Segmental mobility was monitored via the decrease in the calorimetric and dielectric (α relaxation) glass-transition temperatures, Tg and Tg,diel, respectively. The effect was systematic with an increase in the PBSu/PLA ratio, and was rationalized through the plasticizing role of PBSu (low-Tg component) and facilitated also by the simultaneous lowering of the chain length in the copolymers. Dielectric spectroscopy allowed evaluation of the dynamical fragility (cooperativity) of chains, which was strongly suppressed in the copolymers. This finding suggested an increase in free volume or a gradual increase of interchain distances. This phenomenon could favor the natural enzymatic degradation of the systems (compostability), which is limited in neat PLA. We recorded enhancement of nucleation and the crystalline fraction in the copolymers that was likely connected with faster chain diffusion. Further lowering of the Tg with the implementation of crystallization was noted (which seemed a controversial effect) but which indicated crystallization-induced phase separation.