We study Naruse-Newton coefficients, which are obtained from expanding descent polynomials in a Newton basis introduced by Jiradilok and McConville. These coefficients $C_0, C_1, \ldots$ form an integer sequence associated to each finite set of positive integers. For fixed nonnegative integers $a<b$, we examine the set $R_{a, b}$ of all ratios $\frac{C_a}{C_b}$ over finite sets of positive integers. We characterize finite sets for which $\frac{C_a}{C_b}$ is minimized and provide a construction to prove $R_{a, b}$ is unbounded above. We use this construction to obtain results on the closure of $R_{a, b}$. We also examine properties of Naruse-Newton coefficients associated with doubleton sets, such as unimodality and log-concavity. Finally, we find an explicit formula for all ratios $\frac{C_a}{C_b}$ of Naruse-Newton coefficients associated with ribbons of staircase shape.