We extend our recent coarse-grained model describing semicrystalline homopolymers to simulate the morphology and phase transitions of thermoplastic elastomers made of segmented (hard/soft) block copolymers. The generic model is adapted to match the physical characteristics of the two chemical units involved in the copolymer chains by using classic scaling rules. We investigate the crystallization kinetics of the hard segments as well as their phase separation from the soft units in either triblock or pentablock copolymers. We identify the soft segment molecular weight as a key parameter resulting in the following observations when decreasing the temperature from a homogeneous state. On the one hand, the phase separation preceding the crystallization process in triblock copolymers results in a constant temperature of crystallization when varying the soft segment length. On the other hand, the limited phase separation achieved in pentablock copolymers constrains them to crystallize at progressively lower temperatures while increasing the soft segment length. Finally, increasing the soft segment molecular weight was found to lead to a higher relative crystallinity which can be interestingly related to a rise of the loop segment's content.