This is all just my experience, you do what you want.
The oil comes in a little tube, like clipper oil, but for faders.
Deoxit has many products, but there are there are 2 kinds which will probably be most relevant, one is a spray that has a cleaning agent mixed in with the lubricant (fn5s-2 quick evaporating, f5s-h6 extra flushing) But those will still leave the knobs possibly more loose than you want them to be. The one with the cleaner may not be necessary since there is no direct connection to the audio and it doesn’t sound like there’s any issue of conductivity.
The other type is just a lubricant (F100L-L2C) it also comes in a larger size but the product is the same.
You can find these on amazon or any number on places online.
I learned the hard way that although you can use contact cleaner, the fact that it isn’t meant for use on potentiometers makes the extra steps necessary to succeed when using it a bit taxing.
My volume pot was scratchy, I used contact cleaner, then my scratchy pot was no longer scratchy - it was instead loose and wobbly. I tried the lube and it was not sufficient to replace the normal firmness or get rid of the wobble, so I had to go through a labored process to regrease it with the fader grease which is another slightly more expensive product, not just the lube. The grease is thick so without disassembling the pot it’s very difficult to get it into the area between the shaft and the bushing where it is most needed. You would with 100% certainty need to take apart the machine to do it that way.
After application, I had to use various strange methods to push as much as possible down into the gap, all while turning the shaft in order to make sure it was distributing evenly.
When you clean a pot, you’re supposed to take it apart and reapply the grease from the inside, but if you’re going to desolder it to remove from the board and disassemble, go through all that hassle and risk, you may as well just be replacing the pot as it’s almost always a 3 or 4 dollar part.
And it’s easy to say “only spray a little” but in reality, contact cleaner comes out of the can like a fire hose. I’m pretty well acquainted with using spray cans so I’m not exaggerating when I say the contents are under high pressure.
This is what you’re up against, it’s a video I took of what it looks like when you spray “a little” contact cleaner.
So To me, I would start with the $6 fader lube in a squeeze tube, apply a little and turn back and forth a bunch of times to let it seep in, try a little more if needed. Start thinking about spraying things if that doesn’t work.
First I’d remove the knobs and verify the issue still persists and that the hesitation isn’t friction from parts in contact with each other. If it’s internally stiff I’d try what I’ve suggested going one step at a time.
That’s advice though, not the law, so make your own call based on what you’re comfortable doing. Could be better to live with a stiff potentiometer feel.