This paper studies regularity properties of multiplicative stochastic processes on infinite-dimensional Lie groups. We investigate conditions under which these processes admit càdlàg modifications and derive bounds on their local behavior. Our approach builds on the local equivalence of Banach–Lie groups and Banach spaces via the exponential and logarithm, allowing us to transfer analytic estimates and structural results. To illustrate our findings, we consider multiplicative processes on the Heisenberg group.