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I'm currently interviewing with 4 law firms, and am waiting to hear back from about 40 others. Is it OK to ask these 40 law firms to expedite my application, or is this only appropriate to do if I have a pending offer?

Thanks.

Sure, ask them to expedite your application if you want to get dinged

I would wait until you get an offer with an established deadline, and then only contact those of the 40 that you would at least strongly consider over the place at which you had an offer.