Discovering causal relationships from observational data is a challenging task that relies on assumptions connecting statistical quantities to graphical or algebraic causal models. In this work, we focus on widely employed assumptions for causal discovery when objects of interest are (multivariate) groups of random variables rather than individual (univariate) random variables, as is the case in a variety of problems in scientific domains such as climate science or neuroscience. If the group level causal models are derived from partitioning a micro-level model into groups, we explore the relationship between micro- and group level causal discovery assumptions. We investigate the conditions under which assumptions like causal faithfulness hold or fail to hold. Our analysis encompasses graphical causal models that contain cycles and bidirected edges. We also discuss grouped time series causal graphs and variants thereof as special cases of our general theoretical framework. Thereby, we aim to provide researchers with a solid theoretical foundation for the development and application of causal discovery methods for variable groups.