Lauren, I'm afraid I have to begin by saying, "It depends," regarding both relative humidity (RH) and temperature requirements and ranges applied to museum collections; in this case, as applied by lenders of objects to other museums. Some objects are much more delicate than others, and may be damaged by relatively slight changes to their surrounding climate, while objects of ceramic, glass, noble metals and so forth may be little affected; there are always exceptions, of course. Additionally, objects from arid regions may react aversely to what would be considered "average" conditions in a tropical, oceanic region, and vice-versa. Thus, it depends how each object reacts overall, and what its native climate has been.
To answer your question more directly, there may not be much to choose between overall inconsistency and occasional spikes in temperature and RH. Either can have some pretty serious effects on collections objects, and so agencies such as the National Park Service recommends limits for changes in humidity of plus or minus 5% (an overall range of 10%) within a total range of not lower than 40% nor higher than 60% RH. Temperatures for most collection objects are suggested as 59-77 degrees Fahrenheit, kept as constant as possible; if it were not already clear, changes in RH tend to be much more important than changes in temperature. You can view various discussions of this subject by searching websites of the major conservation organizations; the National Park Service information is on: https://www.nps.gov/museum/publications/mhi/chap4.pdf
Best,
Bruce
------------------------------
Bruce MacLeish
Curator Emeritus, Newport Restoration Foundation
Cooperstown NY
------------------------------