  Assessing the applicability of hydrodynamic expansions close to phase
transition points is crucial from either theoretical or phenomenological points
of view. We explore this within the gauge/gravity duality, using the
Einstein-Klein-Gordon model, a bottom-up string theory construction. This model
incorporates a parameter, $B_4$, that simulates different types of phase
transitions in the strongly coupled field theory existing at the boundary. We
thoroughly examine the thermodynamics and dynamics of time-dependent,
linearized perturbations in the spin-2, spin-1, and spin-0 sectors. Our
findings suggest that "hydrodynamic series breakdown near transition points" is
valid exclusively for second-order phase transitions, not for crossovers or
first-order phase transitions. Additionally, we observe that the
high-temperature and low-temperature limits of the radius of convergence for
the hydrodynamic series ($q^2_c$) are equal. We also discover that the
relationship $(\text{Max}\vert q^2_c \vert)_{\text{spin-2}} < (\text{Max}\vert
q^2_c\vert)_{\text{spin-0}} < (\text{Max}\vert q^2_c \vert)_{\text{spin-1}}$ is
consistent for different spin sectors, regardless of the phase transition type.
At the chaos point, we observe the emergence of pole-skipping behavior for both
gravity and scalar perturbations at $\omega_n = - 2\pi T n i$. Lastly,
comparing the chaos momentum with $q^2_c$, we find that $q^2_{ps} < q^2_c$,
except for extremely high temperatures.
