Structural causal models (SCMs) are widely used in various disciplines to represent causal relationships among variables in complex systems. Unfortunately, the true underlying directed acyclic graph (DAG) structure is often unknown, and determining it from observational or interventional data remains a challenging task. However, in many situations, the end goal is to identify changes (shifts) in causal mechanisms between related SCMs rather than recovering the entire underlying DAG structure. This paper focuses on identifying mechanism shifts in two or more related SCMs over the same set of variables---without estimating the entire DAG structure of each SCM.In this work we assume that each SCM belongs to the class of nonlinear additive noise models. We prove a surprising result where the Jacobian of the score function for the mixture distribution reveals information about shifts in general non-parametric functional mechanisms. Once the shifted variables are identified, we leverage recent work to estimate the structural differences (if any) for the shifted variables. Experiments on synthetic and real-world data are provided.