Legally, inherited guns do not need to be transferred through an FFL.
Shipping can be problematic, but long guns may be shipped via USPS, UPS Ground, and Fedex Ground, and federal law says that guns NOT going to a license holder must be declared to the shipper, so in practicality, you may find that they must be sent to FFL.
If that turns out to be the only option, you definitely need to find one that will either just charge you a fee for handling the shipment, or if they insist on actually doing 4473 transfer, one that will do them all for one fee rather than charge for each. Perhaps a notarized document of some sort could let an FFL not do 4473 but simply charge for handling the shipment? I dunno, ask around.
You might also look into having them sent by a moving company, all in one or two crates; last time I did cursory look into this, long guns are okay, but you'll need to insure them for all they are worth, of course.
- OS