This paper presents a novel method for computing certain particle locations in the multi-species $q$-TAZRP (totally asymmetric zero range process). The method is based on a decomposition of the process into its discrete-time embedded Markov chain, which is described more generally as a monotone process on a graded partially ordered set; and an independent family of exponential random variables. A further ingredient is explicit contour integral formulas for the transition probabilities of the $q$-TAZRP. The main result of this method is a shift invariance for the multi-species $q$-TAZRP on the infinite line. By a previously known Markov duality result, these particle locations are the same as joint $q$-moments. One particular special case is that for step initial conditions, ordered multi-point joint $q$-moments of the $n$-species $q$-TAZRP match the $n$-point joint $q$-moments of the single-species $q$-TAZRP. Thus, we conjecture that the Airy$_2$ process describes the joint multi-point fluctuations of multi-species $q$-TAZRP. As a probabilistic application of this result, we find explicit contour integral formulas for the joint $q$-moments of the multi-species $q$-TAZRP in the diffusive scaling regime.