Thermally stimulated depolarization currents, TSDC, experiments have been performed on a series of poly(styrene)-$b$-poly(butadiene)-$b$-poly$(\ensuremath{\epsilon}$-caprolactone) triblock copolymers SBC with different proportions of the poly$(\ensuremath{\epsilon}$-caprolactone) crystallizable block, PCL. The morphology of the segregated microphases varies with the PCL content and has been observed by transmission electron microscopy. The crystallinity of the PCL block is estimated by wide angle x-ray scattering, WAXS. The relaxation times distribution is extracted by a numerical decomposition of the TSDC spectra and it is shown that this distribution is not significantly changed on going from the homopolymer to the triblock copolymer with 16 wt % to 77 wt % of PCL in the original samples. Better segregation of the mesophase structure is reached when the samples are annealed at 413 K and important variations in the TSDC and WAXS spectra are observed as a result of the thermal treatment. For the ${\mathrm{S}}_{09}{\mathrm{B}}_{14}{\mathrm{C}}_{77}$ triblock copolymer the results obtained can be explained by postulating the existence of a rigid amorphous phase in the PCL block. Such rigid amorphous phase is located between the core-shell cylinders formed by the other blocks [with poly(styrene)(PS) as core and poly(butadiene)(PB) as shell] and is constrained by undulated lamellae of crystalline PCL material. In the case of ${\mathrm{S}}_{35}{\mathrm{B}}_{15}{\mathrm{C}}_{50}$ triblock copolymer, an important amount of diffuse PS-PCL interphase where the homopolymers are mixed must be present before annealing. The results for the material with the less abundant PCL block are explained as a result of the confinement in nanotubes of PCL surrounded by PB embedded in a vitreous PS matrix. Broadband dielectric experiments on these same materials confirm the results obtained by TSDC spectroscopy.