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I establish the dynamical core--corona initialization framework (DCCI2) as a state-of-the-art dynamical framework that is capable of describing small and large colliding systems at the LHC energies. Under the core--corona picture, contributions from both equilibrated (core) and non-equilibrated (corona) components are implemented. I describe the dynamical separation of the system into the core and corona at the initial stage by incorporating the core--corona picture into the novel dynamical initialization framework. With DCCI2, I simulate $p$+$p$ collisions at $\sqrt{s}=7, 13$ TeV and $Pb$+$Pb$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV. Especially, I extract the fractions of core and corona components in final hadron yields in $p$+$p$ and $Pb$+$Pb$ collisions as functions of multiplicity, and reveal that the core components become dominant at $\langle dN_{\mathrm{ch}}/dη\rangle_{|η|<0.5} \approx 20$. I also find that the corona contribution at very low $p_T$ (below $p_T\approx1$ GeV) is non-negligible even in $Pb$+$Pb$ collisions and show that such contributions significantly affect $p_T$-integrated flow coefficients. These results strongly suggest the importance of considering non-equilibrated components to extract transport coefficients of quark-gluon plasma from model-to-data comparisons quantitatively.