The crossing of the Galactic disc by a globular cluster (GC) could produce star formation due to gravitational focusing or compression of disc material. We report on simulations of the effect on disc material which reveal that the crossing can sometimes cause local gravitational focusing of disc material. We also present the salient points of a little-known paper by Levy, which shows that strong compression can result from the shock wave generated by GC disc crossing. The main thrust of our paper is a search for remnants of disc crossings by GCs. Using the gravitational potential of the Galaxy to locate the position of the most recent crossings of a subset of fifty-four GCs reveals that systematic errors and uncertainties in initial conditions limit the scope for unequivocal identification. From the subset of fifty-four, six possible search sites with the best constraints are retained for further scrutiny. Three of the six potentially promising search areas in the disc are from GCs NGC 3201, 6397 and 6838, for which we cannot rule out some observed star associations observed nearby as being remnants. The other three of the six areas are too large to provide meaningful identification of remnants. Also, a possible remnant (open cluster NGC 6231) is shown not to be due to GC impact, contrary to a previous report. In a more wide-ranging screening of 155 GCs, we identify which GCs are compatible with being responsible for the formation of any of the Galaxy's five most prominent star superclusters.