Summary: We compare the solutions of one-scale Dyson-Schwinger equations in the Minimal subtraction (MS) scheme to the solutions in kinematic (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counter terms in perturbation theory. As concrete examples, we examine three different one-scale Dyson-Schwinger equations, one based on the D=4 multiedge graph, one for the D=6 multiedge graph and one mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson-Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact non-perturbative solution to one of the non-linear DSEs of the toy model.