In addition to what Armorer-At-Law said...
If the gun in question is a long gun, i.e. rifle or shotgun, the transaction may legally take place at an FFL in any
state provided that
the transaction is legal in both the current owner's and intended recipient's states of residence. In the case of TX, there is no law against it, although I am not sure about AR. A phone call to an AR FFL located near the TX border will probably put the question to rest.
FWIW if you decide to check AR state law yourself, be aware that federal law used
to disallow adjacent-state transfers by default unless both states passed laws allowing them. Many states subsequently passed laws allowing adjacent-state transfers, but these laws are often worded in confusing double negatives, e.g. "...long gun transfers from duly licensed TX dealers are not
prohibited". These statutes are now legal dead weight because the federal restriction was removed long ago, but they remain on the books in many places. IOW if you encounter a law that seems to prohibit a transfer through a TX FFL, read the "nots" carefully, it may not say what you initially think.
OTOH handguns must be transferred through an FFL in the intended recipient's state of residence, period.
Expect to pay a $10-$50 fee per firearm for the transfer. $20-$30 is typical.
If your cousin decides to ship to a TX FFL (handgun or not), make sure to check beforehand whether the FFL will accept the shipment. Although federal law allows FFLs to accept handguns from non-FFLs, it doesn't require
them to do it. Some FFLs choose not to accept shipments from non-FFLs because they don't want to be stuck holding the bag if the gun turns out to be stolen or illegally modified, or deal with the consequences if the shipper's identifying information is missing, illegible, or incomplete (the receiving FFL is legally required to record this).
- If a gun is over 50 years old or appears on the ATF's Curio & Relics list, a Curio & Relic FFL holder (aka 03 FFL, C&R licensee, or licensed collector) may receive it from a resident of any state provided that the transaction is legal in both the current owner's and intended recipient's states of residence. You can apply for a C&R FFL yourself; it's intended for individuals, and most people who are eligible to purchase a firearm through a standard background check can get one. You also get discounts from mail-order shooting supply houses, which will often offset the $30 license fee in a hurry. Search the Curio & Relic forum for more info.
- Legal antiques do not require FFL transfers. The most commonly encountered types of legal antiques are black powder muzzleloaders, black powder replicas that don't accept metallic cartridges, and firearms made before 1/1/1899.
What I've written above does not necessarily apply to firearms subject to the National Firearms Act of 1934 (NFA). They are subject to many additional restrictions.