If I were to buy an item from the duty free that was too big to carry on and needed to go into the hold, what is the procedure?
Oversize carry on baggage is often tagged and sent to the hold, however if you purchase something in duty free that will not conform to your contractual carry on limits, the airline is under no obligation to do so.
A case of
caveat emptor, in that your purchase is little more than a contract between you and the retailer (duty free shop.) It has nothing to do with your contract between you and the air carrier, nor does it modify the latter in any implied way.
You might normally expect common sense to apply, however if the passenger attempts to carry on an article that will not comply with the normal carry on rules, or is likely to involve the carrier in extra cost or inconvenience, then the carrier would have the right to refuse carriage of said item, or to charge the passenger for the additional item, at its absolute discretion.