Is it possible to mix the rings in the crankset?
I'm going to answer with regard to putting smaller-than-stock small rings on a triple crank in order to get lower climbing gears. Doing it with the middle and large rings presents other challenges.

It's possible, and a lot of people ride setups like this. Here are some hitches you may encounter:


You may, and often will, exceed the rear derailer's 'Total Capacity' number. One of a rear derailer's jobs is to keep the chain tensioned over all the gear combinations. Total Capacity defines how wide of a gear range a given derailer is able to do this for. What this means in practice is that when the chain is long enough to cover the large/large combination safely, which it always should be, it will be slack in some number of the small/small combinations. Riding in these gears introduces a risk of derailment or jamming since the chain is more free to bounce around. A lot of cyclists decide they're fine with that risk and just stay out of those gears, which works well with the habits of most touring and utilitarian cyclists in particular since for them the small ring is mostly just for steep climbs. The exact same concern arises when you try to get your lower gears by putting on a bigger cassette, only in that case you also need to lengthen the chain.
Current (I'd say 'modern' but I hold out hope for the pendulum swinging back the other way someday) triple front derailers can have pretty specific chain contact points sculpted into them, and they correspond with the ring combinations they're designed to be run with. And middle chainrings have shift-assisting features (ramps and pins) that expect the chain to come off a specific inner ring. When people talk about how getting smooth front indexed shifting relies on all the components being designed to work as a system, these are some major pieces of that. You will tend to lose some shifting performance for these reasons if you put on a smaller inner ring. Usually it will be in the realm of still working, but just being more sluggish, clunky, and finicky to adjust and keep adjusted. It is definitely something of an experiment every time, which is a reason shops may be reluctant or unwilling to do it.
Friction front shifting handles this kind of setup much better because you have more freedom to finesse it. STI in particular is clumsy to intentionally overshift. It can be easy for performance to feel mediocre or compromised when you do this on a bike with STI.
I don't know all the reasons, but bikes set up this way are often more prone to chain drop on the downshifts. Using a simple plastic chain catcher of the dog-tooth variety is a good idea and usually solves the problem more or less completely.