This understanding is primarily for the Airbus. Please correct if incorrect.
GPS feeds to IRS. GPS->IRS=GPIRS. This is higher priority than radio updating.
GPIRS feeds to FM for position update.
3 IRS makes MIX IRS. [GPIRS/radio update]-[MIX IRS]=BIAS.
In case of GPIRS/radio update failure: MIX IRS+BIAS feeds to FM
Position is fed to FM for navigational use.