What voltage/current is applied by modern multimeters to determine resistance?
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That will vary by multimeter but most professional level ones will include that information in the datasheet and it can vary by range. Here's the table from the documentation for an Agilent 34461A:
Using that I just measured a cheap "no name" meter and it had a maximum 320 mV voltage across all ranges. So hopefully that gives you a rough idea, and you could find further information in the datasheets for the better brands of meters.
Electronic multimeters are one thing but an analog multimeter like a Simpson 260 can put out a lot of current when measuring low values of resistance. That is why they usually use a D battery for the low ohms ranges. The current can be in the hundreds of milliamperes on the X1 range.