We study the transient phenomena appearing in a subgap region of the double quantum dot coupled in series between the superconducting and normal metallic leads, focusing on the development of the superconducting proximity effect. For the uncorrelated nanostructure we derive explicit expressions of the time-dependent occupancies in both quantum dots, charge currents, and electron pairing induced on individual dots and between them. We show that the initial configurations substantially affect the dynamical processes, in which the in-gap bound states emerge upon coupling the double quantum dot to superconducting reservoir. In particular, the superconducting proximity effect would be temporarily blocked whenever the quantum dots are initially singly occupied. Such {it triplet}/{it Andreev blockade} has been recently reported experimentally for double quantum dots embedded in the Josephson [D. Bouman et al., Phys. Rev. B 102, 220505 (2020)] and Andreev [P. Zhang et al., arXiv:2102.03283 (2021)] junctions. We also address the role of correlation effects within the lowest-order decoupling scheme and by the time-dependent numerical renormalization group calculations. Competition of the repulsive Coulomb interactions with the superconducting proximity effect leads to renormalization of the in-gap quasiparticles, speeding up the quantum oscillations and narrowing a region of transient phenomena, whereas the dynamical Andreev blockade is well pronounced in the weak inter-dot coupling limit. We propose feasible methods for detecting the characteristic time-scales that could be observable by the Andreev spectroscopy.
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