We explore model-independent constraints on the Universe kinematics up to the snap and jerk hierarchical terms, considering the latest baryon acoustic oscillation (BAO) release provided by the DESI collaboration. We intend to place novel and more stringent constraints on the cosmographic series, incorporating three combinations of data catalogs: the first made by BAO and observational cosmic chronometers, the second made by BAO and type Ia supernovae, and the last including all the cited data sets. Considering the latest BAO data provided by the DESI collaboration and tackling the $r_d$ parameter to span within the range $ $ Mpc, with a fixed step of $ r_d=2$ Mpc, we employed Monte Carlo Markov chain analyses based on the Metropolis algorithm to fix novel bounds on the cosmographic series, fixing the deceleration, $q_0$, the jerk, $j_0$, and the snap, $s_0$, parameters, up to the $2$sigma level. A comparison between the results of the Planck satellite with those obtained by the DESI collaboration is also reported Our findings showcase a significant departure in terms of $j_0$ even at the $1$sigma confidence level, albeit compatible with the Lambda CDM paradigm in regard to $q_0$ and $s_0$ at the $2$sigma level. Analogously, the $h_0$ tension appears alleviated in the second hierarchy when including snap. Our method excludes models that significantly depart from the standard cosmological model. Particularly, direct comparisons with the Lambda CDM and $w$CDM models and the Chevallier-Polarski-Linder parameterisation are explored, which definitively favour the wCDM scenario over other approaches, contradicting the findings of the original DESI collaboration.