Does cosmological expansion affect local physics?

In this contribution I wish to address the question whether, and how, the global cosmological expansion influences local physics. I argue that a pseudo Newtonian picture can be quite accurate if "expansion" is taken to be an attribute of the inertial structure rather than of "space" in some substantivalist sense. This contradicts the often-heard suggestion to imagine cosmological expansion as that of "space itself". Regarding General Relativity, I emphasise the need for proper geometric characterisations in order to meaningfully compare localised systems in different spacetimes, like black holes in static and expanding environments. Examples of this sort are discussed in some detail to clearly map out the problems.