In the present study, a fully plant-based sustainable copolyester series, namely poly(butylene 2,5-furandicarboxylate)-block-poly(caprolactone)s (PBF-block-PCL)s were successfully synthesized by melt polycondensation combining butylene 2,5-furandicarboxylate with polycaprolactone diol (PCL) at different weight ratios. Differential scanning calorimetry (DSC) showed that only PBF underwent melting, crystallization from the melt, and cold crystallization. Thermogravimetric analysis (TGA) revealed outstanding thermal stability, exceeding 305 °C, with further improvement in thermal and thermo-oxidative stability with increasing PCL content. Broadband dielectric spectroscopy (BDS) revealed that at low temperatures, below the glass transition (Tg) all copolyesters exhibited two relaxation processes (β1 and β2), whereas the homopolymer PBF exhibited a single β-relaxation, which is associated with local dynamics of the different chemical bonds present in the polymer chain. Additionally, it was proved that an increase in PCL content affected the dynamics of the chain making it more flexible, thus providing an increase in the value of the room temperature free volume fractions (fv) and the value of elongation at break. These effects are accompanied by a decrease in hardness, Young's modulus, and tensile strength. The described synthesis enables a facile approach to obtain novel fully multiblock biobased copolyesters based on PBF and PCL polyesters with potential industrial implementation capabilities.