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Recently \citet{PEDE:2019ApJ} proposed a phenomenologically emergent dark energy (PEDE) which consider that the dark energy density evolves as $\widetildeΩ_{\rm{DE}}(z)\,=\,Ω_{\rm{DE,0}}\left[ 1 - {\rm{tanh}}\left( {\log}_{10}(1+z) \right) \right]$ with the advantage that it does not have degree of freedom. Later on, \citet{PEDE:2020} proposed a generalized model by adding one degree of freedom to the PEDE model, encoded in the parameter $Δ$. Motivated by these proposals, we constrain the parameter space ($h,Ω_m$) and ($h,Ω_m, Δ$) for PEDE and Generalized Emergent Dark Energy (GEDE) respectively, by employing the most recent observational (non-) homogeneous Hubble data. Additionally, we reconstruct the deceleration and jerk parameters and estimate yield values at $z=0$ of $q_0 = -0.784^{+0.028}_{-0.027}$ and $j_0 = 1.241^{+0.164}_{-0.149}$ for PEDE and $q_0 = -0.730^{+0.059}_{-0.067}$ and $j_0 = 1.293^{+0.194}_{-0.187}$ for GEDE using the homogeneous sample. We report values on the deceleration-acceleration transition redshift with those reported in the literature within $2σ$ CL. Furthermore, we perform a stability analysis of the PEDE and GEDE models to study the global evolution of the Universe around their critical points. Although the PEDE and GEDE dynamics are similar to the standard model, our stability analysis indicates that in both models there is an accelerated phase at early epochs of the Universe.