This study explores model-independent constraints on the kinematics of the universe up to the snap and jerk hierarchical terms using the latest Baryon Acoustic Oscillation (BAO) data from the DESI collaboration. The $r_d$ parameter, associated with BAO data, is fixed within the range [144, 152] Mpc with a step of 2 Mpc, aligning with Planck and DESI results. Using Monte Carlo Markov chain analyses, the cosmographic series are constrained, incorporating three combinations of data catalogs: BAO with observational Hubble data, BAO with type Ia supernovae, and all three datasets combined. The results show significant constraints on the deceleration $q_0$, jerk $j_0$, and snap $s_0$ parameters at the 2-$\sigma$ level, with a notable departure in $j_0$ even at the 1-$\sigma$ confidence level. The $h_0$ tension is alleviated in the second hierarchy, including the snap. A comparison with the $\Lambda$CDM, $w$CDM, and Chevallier-Polarski-Linder (CPL) models reveals that the $w$CDM scenario is better adapted to the BAO data, showing a severe deviation from the standard cosmological model in the jerk parameter. The study concludes that further data release is needed to clarify the numerical tensions in $j_0$ and $q_0$.This study explores model-independent constraints on the kinematics of the universe up to the snap and jerk hierarchical terms using the latest Baryon Acoustic Oscillation (BAO) data from the DESI collaboration. The $r_d$ parameter, associated with BAO data, is fixed within the range [144, 152] Mpc with a step of 2 Mpc, aligning with Planck and DESI results. Using Monte Carlo Markov chain analyses, the cosmographic series are constrained, incorporating three combinations of data catalogs: BAO with observational Hubble data, BAO with type Ia supernovae, and all three datasets combined. The results show significant constraints on the deceleration $q_0$, jerk $j_0$, and snap $s_0$ parameters at the 2-$\sigma$ level, with a notable departure in $j_0$ even at the 1-$\sigma$ confidence level. The $h_0$ tension is alleviated in the second hierarchy, including the snap. A comparison with the $\Lambda$CDM, $w$CDM, and Chevallier-Polarski-Linder (CPL) models reveals that the $w$CDM scenario is better adapted to the BAO data, showing a severe deviation from the standard cosmological model in the jerk parameter. The study concludes that further data release is needed to clarify the numerical tensions in $j_0$ and $q_0$.