In this work, we study a series of sustainable block copolymers based on polylactide, PLA, and poly(propylene adipate), PPAd, both polymers being prepared from renewable resources. Envisaging a wide range of future applications in the frame of a green and circular economy, e.g., packaging materials replacing conventional petrochemicals, the employment of PPAd aims at lowering the glass transition and melting temperatures of PLA and, finally, facilitation of the enzymatic degradation and compostability. The copolymers have been synthesized via ring opening polymerization of lactides in the presence of propylene adipate oligomers (5, 15 and 25%). The direct effects on the molecular mobility by the structure/composition are assessed in the amorphous state employing broadband dielectric spectroscopy (BDS) and calorimetry. BDS allowed the recording of local PLA and PPAd dynamics in all cases. The effects on local relaxations suggest favoring of interchain interactions, both PLA-PPAd and PPAd-PPAd. Regarding the more important segmental dynamics, the presence of PPAd leads to faster polymer chain diffusion, as monitored by the significant lowering of the dielectric and calorimetric glass transition temperature, Tg. This suggests the plasticizing role of PPAd on PLA (majority) in combination with the lowering of the average molar mass, Mn, in the copolymers from ∼75 to ∼30 kg mol-1, which is the actual scope for the synthesis of these materials. Interestingly, a strong suppression in fragility (chain cooperativity) is additionally recorded. In contrast to calorimetry and due to the high resolving power of BDS, for the higher PPAd fraction, the weak segmental relaxation of PPAd was additionally recorded. Overall, the recordings suggest a strong increase in free volume and two individual dynamic states, one for 0 and 5% PPAd and another for 15 and 25% PPAd. Within the latter, we gained indications for partial phase nano-separation of PPAd. Regarding indirect effects, these were followed via crystallization. Independent of the method of crystallization, namely, melt or cold, the presence of PPAd led to the systematic lowering of crystallization and melting temperatures and enthalpies. The effects reflect the decrease of crystalline nuclei, which is confirmed by optical microscopy as in the copolymers fewer although larger crystals are formed.