The fragmentation equation is a linear integro-differential equation widely used in a variety of fields to characterize the dynamics of a population’s density. Moments of the solution are frequently considered, and the development of numerical schemes able to preserve their dynamics is a crucial issue. In this work, we review the properties of an existing weighted finite volume scheme introduced in, aiming at preserving both the zeroth (total number density) and first (total mass) order moments of the solution over time. A drift in the discrete zeroth-order moment dynamics is quantified. Moreover, the expected second-order convergence is not always attained, and a new consistency proof is given for uniform meshes, together with conditions linking both the fragmentation rate and kernel with the consistency order. The result is illustrated on a typical self-similar fragmentation kernel together with monomial fragmentation rates, showing how some configurations yield a degraded convergence order. This work allows to further characterize the weighted numerical scheme regarding its convergence, and to discuss the moments-preserving property of finite volume schemes.