My understanding is that the FMS takes account of the IRS position and the possible circle of uncertainty caused by the system drift. It compares this to DME/DME crosscuts with their much smaller potential error and computes a likely position between the two but heavily biased to the DME/DME position, ideally within all the circles and ranges of uncertainty. If the double DME crosscut is lost it assumes the position is the IRS position with the last known bias applied to it. This means, in short, that the INS/IRS is just allowed to drift and the FMS calculates its position on the basis of all available information.
Which IRS/INS position is taken depends on type. The L1011 had three INS and the FMS 'triple mixed' the positions of the three systems to arrive at an averaged INS position. I understand the B737 uses only the left hand IRS as an FMS position input under normal circumstances.
The old INS could be position updated in flight, I don't know about IRS. You used to select POS and HOLD coming up to a beacon overhead, insert the lat and long of the beacon, and release the HOLD button when overhead.
Finally, the radio updates into the FMS vary. The B737 only uses DME/DME, some systems will also use DME/VOR and a few very old systems will also use VOR/VOR.