Galactic cosmic ray (GCR) flux is modulated by both particle drift patterns and solar wind structures on a range of timescales. Over solar cycles, GCR flux varies as a function of the total open solar magnetic flux and the latitudinal extent of the heliospheric current sheet. Over hours, drops of a few percent in near-Earth GCR flux (Forbush decreases, FDs) are well known to be associated with the near-Earth passage of solar wind structures resulting from corotating interaction regions (CIRs) and transient coronal mass ejections (CMEs). We report on four FDs seen at ground-based neutron monitors which cannot be immediately associated with significant structures in the local solar wind. Similarly, there are significant near-Earth structures which do not produce any corresponding GCR variation. Three of the FDs are during the STEREO era, enabling in situ and remote observations from three well-separated heliospheric locations. Extremely large CMEs passed the STEREO-A spacecraft, which was behind the West limb of the Sun, approximately 2–3 days before each near-Earth FD. Solar wind simulations suggest that the CMEs combined with pre-existing CIRs, enhancing the pre-existing barriers to GCR propagation. Thus these observations provide strong evidence for the modulation of GCR flux by remote solar wind structures.