As Far As The Law Is Concerned:
Once they get The Master—Compact Disc / WAV File / MP3 File / etc. — which is what they paid for that day; you are not obligated to hold on to the multi-tracks. After they paid and got their product that day you are free to erase the files, if you wish—if they didn’t pay, you have every right to erase the files immediately. If you want to continue, that is up to you (unless you have a written contract that states otherwise). 
It is That Master—Compact Disc, WAV, MP3, or other audio file format—that they paid for at the end of the day, which they got. Right? 
That’s usually how it works…
1. They pay.
2. They get a master copy (whether they consider it completed or not—that is subjective).
…and That’s It. 
If the artist claims that the tracking was incomplete, this is a subjective statement, because we are dealing with art—the artist could say that forever and force you to go bankrupt. If the artist could use that as an excuse to get something from you that does not belong to them, you would have no property rights. The moment that someone breaches a contract, that contract is no longer valid.
Sometimes you might enter in to a weeks/months long set of tracking sessions.
Unless you made a contractual agreement and they pre-paid for it, you do not have to keep it. 
This gives you a lot of power, because the multi-tracks are usually store(d) on your equipment, not theirs. 
If you choose to continue tracking another day, or provide the multi-tracks, that is up to you—not them. (unless you are storing those multi-tracks on their hardware—not yours—as you record them—in which case you would only be obligated to hand over their hardware) 
If they pre-paid with a contractual agreement, then you may be required to hold onto it—or at least continue recording—until the pre-paid time has been expended.
However, if they breach the contract by not paying, or leaving to have it completed elsewhere, the contract is broken, and you are back in control—with no obligations. You do NOT need to provide a refund, or keep the files (unless a signed contract states otherwise). 
If they ask for the multi-tracks and you say yes, you are only obligated to provide those multi-tracks if they pay for that time and storage (giving you something to put them on)—you aren’t backing out if you do not provide the multi-tracks because they did not pay, they are backing out—not you. Even if you did say you would provide them, if they did not pay in full, you are not obligated to provide them (because you were making a statement based on the conclusion that they would pay). 
Some people think that studios are required to store multi-tracks forever (it’s a mystery as to why anyone would think something so absurd), but their is no legal precedent for something as absurd as this what-so-ever—and you won’t find that precedent for a darn good reason. 
No sane judge would rule that a studio be re-defined as a storage center.
You can check the law books, there is no prior court ruling to use as precedent.
Clients in this business don’t usually pay monthly fees for storage of their multi-tracks. Tell them that they aren’t paying storage fees if they get on your case, and remind them that they got what they paid for already—it’s your choice (your right) to choose whether to continue recording them or not.
When you record someone, you hold the SR Copyright until the recording is paid in full (not the PA copyright—unless you produced it as well). 
If your client sounds like they might want the multi-tracks, and you are willing to give them up, you should advise your client to bring a USB storage device so you can give them the multi-tracks before they leave.
If you would rather not provide multi-tracks to your client, you are neither required to provided the multi-tracks either—you are not legally obligated to do so. 
The only product you are providing is a stereo mix-down/master, unless stated otherwise. 
You are an audio engineer, not a storage center, it’s important to know your rights!
Know Your Rights!
1. Avid Pro Audio Community (Avid Pro Audio Community RSS)
“He who pays the studio time, owns the product”. “The hard drive that stores both the sessions/projects and the [stereo] Mix-down can be declared as the “Master”. “…No payment received/in full for services rendered, the ownership of the “Masters” stays with the financier, or studio owner.“
2. Audio mastering (Wikipedia)
“The term “master” is commonly used to describe the finalized product used for duplication.”
3. HFA – Find Out – Faq (HFA – Find Out – Faq)
“Master use rights can only be obtained from the owner of the master recording.”
4. Copyright Law of the United States of America (U.S. Copyright Office)
“Ownership of a copyright, or of any of the exclusive rights under a copyright, is distinct from ownership of any material object in which the work is embodied.”
5. Copyright Circular 56
“Copyright in a sound recording protects the particular series of sounds that are “fixed” or embodied in a recording against unauthorized reproduction and revision.”
6. Mechanic’s lien (Wikipedia)
“The lien was created by operation of law by the fact of the artisan working on the personal property item or attaching additional material to it.”