The crystallization kinetics and crystallization-induced morphological formation of an asymmetric double-crystalline block copolymer, syndiotactic polypropylene-block-poly(epsilon-caprolactone) (sPP-b-PCL), have been investigated by time-resolved simultaneous small-angle and wide-angle X-ray scattering (SAXS/WAXS). The sPP-b-PCL under study exhibited hexagonally packed cylinder morphology in the melt state, where the minority sPP block formed the cylindrical microdomains dispersed in the PCL matrix. The crystallization behavior was studied by imposing two types of crystallization histories: (1) two-stage crystallization, where the diblock was first cooled to the temperature T-c(sPP) situating between the melting points of the two components (T-m(PCL) < T-c(sPP) < T-m(sPP)) to allow sPP crystallization to saturation followed by cooling to T-c(PCL) < T-m(PCL) to induce PCL crystallization; (2) one-stage crystallization, where the system was cooled directly to T-c < T-c(PCL) to allow the two components to crystallize competitively. In both cases, the crystallization of sPP block was in general able to disrupt the melt structure and transformed it into a crystalline lamellar morphology. For the two-stage crystallization process, the PCL block was found to exhibit a faster crystallization at a given T-c(PCL) when the sPP block was precrystallized at higher T-c(sPP). This "interactive crystallization kinetics" was attributed to the mediation of the stretching of PCL blocks by the thickness of sPP crystalline domains which depended on T-c(sPP). In the one-stage process, the crystallization events of the two blocks became more competitive with decreasing T-c. The morphological perturbation induced by crystallization was also more hindered at lower T-c, such that a significant portion of sPP blocks remained confined within the cylindrical microdomains so as to suppress the sPP crystallinity.