Mantissa distributions generated by dynamical processes continue to attract much interest. In this article, it is demonstrated that one-dimensional projections of (at least) almost all orbits of many multi-dimensional nonautonomous dynamical systems exhibit a mantissa distribution that is a convex combination of a trivial point mass and Benford's Law, i.e. the mantissa distribution of the non-trivial part of the orbit is asymptotically logarithmic, typically for all bases. Both linear and power-like systems are considered, and Benford behaviour is found to be ubiquitous for either class. The results unify previously known facts and extend them to the nonautonomous setting, with many of the conclusions being best possible in general.