Thanks for digging those up, I was just looking at the datasheets, should have gone deeper. It's frustrating that continuity check is usually presented as a checkbox feature rather than one worthy of specs (I, V, latch speed/hold).
Fluke 87V (8V/1mA): agreed, continuity voltage/current are one of the worst I've seen for in-situ prodding, probably a holdover from an earlier era.
17B+ (0.53V/250uA): appears to not state any precise values for continuity check in the manual (just N/A, N/A, N/A, very frustrating). But in practice I saw in a review and can confirm on my own unit that these values are true.
KY1012: 0.5V/0.4mA is quite acceptable as well.
DT4252: 1.8V lags behind, that's plenty to start turning on protection diodes, transistors, and such.