I think right off the top there is no generic "how to maintain a disconnect" other than general maintenance standards like NFPA 70B or NETA MTS. Breakers are a different story. EVERY MCCB manufacturer refers you to NEMA AB-4 as the reference standard for MCCB maintenance.
Going by NEMA AB-4, it is very, very simple. If the breaker trips, do a 10 second visual inspection. NEMA AB-4 is completely free to read and download (PDF) so I recommend doing it. They have some very nice color pictures explaining what to look for that most maintenance people can easily follow. If you see anything bad, replace the breaker. It's that simple. Next once a year, exercise it (open and close it 2 or 3 times) and again, do the visual inspection. Next AB-4 talks about doing breaker testing but they don't exactly tell you how to do that or how often, only that it is a good idea to do it.
Switching to disconnects, if the contact tips are visible (disconnects only) it is a fairly easy visual inspection to check for excessive contact tip wear. If the contact tip is worn down to the point where there isn't a "shoulder" anymore to where it is going to start eating into the copper bus supporting it, the tip is worn out and needs replacing. On vacuum interrupters you close the interrupter then there is a way to measure the plunger on the bottle to determine how much contact tip is left. When it wears down too far, replace it. No way to do this on MCCB's because you can't see inside. They need to take a note from the vacuum interrupter market.
As an example of what to look for on disconnects, frequently on medium voltage equipment is that with a lot of it you can get condensation on the top of the enclosure that eventually turns into bubbled up paint and rust. The rust drips with condensation in outdoor enclosures. Usually according to Murphy's Law and depending on who designed it, it always lands on the center insulator and eventually causes a flash over and destroys the disconnect. This tends not to be a problem with Federal Pacific Autojets or the Joslyn-Clark style design but tends to be a common failure mode in a lot of others. So even that little rust on the enclosure that you may think is OK is a good idea to clean up and repaint, especially at 4160 and above where cleanliness is key to reliability.
Then some disconnects have a requirement to lubricate the bearings or some sliding surfaces. This varies from one manufacturer to another. It is in the 1 or 2 paragraphs in the instructions if there is anything at all. Since it seems like you are focussed more on the 600 V class 30-600 A disconnects, there isn't anything at all. These days they have gotten so cheap that usually you don't even replace contact tips...you replace it all. I've also run into some AB disconnects in MCC that were way out of alignment and had to be adjusted. This happens as cabinetry and equipment settles over time. Again it's mostly visual inspections.
So getting into testing proper, this is approximately how we do it and how most customer standards ask for it. First do insulation resistance on everything to check for failed insulation. Looking for usually 100 Megaohms or more. With vacuum interrupters it is done slightly differently if they want it done. Most of the instructions call for using a hi pot essentially as a high voltage megger that is supposed to detect failed vacuum in a bottle. The trouble is that it ALWAYS passes or you'd know the bottle is failed anyways. There is a tester on the market that actually tests the vacuum in place. It is fairly expensive but very nice because it also tells you how much life is left on your bottle.
Next step is to close the contacts and measure resistance of the contacts using a micro-ohm meter. What you are looking for is that they are generally under 1 milliohm and that the readings are within 50% of the average. There are some more refined values to use but generally speaking what we are looking for is either very severe contact damage (pitting) or more likely spring pressure not holding the contact tips together sufficiently.
On breakers the next step is to test the trip unit. Essentially what we are doing is applying a current to the breaker and simulating trip conditions, then timing how long it takes for the breaker contacts to actually open. We are not testing a new breaker design, just testing each trip function to verify that it is working properly. There are two ways of doing this, primary and secondary injection. In primary injection we have a large test set. They usually weigh several hundred pounds. We apply a voltage of around 1 to 10 V but current that is thousands of amps. The current that we apply usually tests 2 points on the thermal curve and a point below and above the instantaneous curve. We are comparing the test results to the published trip curves for the breaker itself. Any deviation from this indicates a breaker out of calibration. With secondary injection the breaker will have a little port on it some place where we can plug in a box that does the same kind of test except that instead of applying full current to it, we are using control circuit voltages and injecting test currents on the other side of the current sensor of the breaker. Instead of weighing hundreds of pounds and truly putting some stress on the breaker itself, the tester weighs maybe 10 pounds and fits in a backpack. But there is no standard for these. Each manufacturer has between 1 and maybe 5 different testers. They are all priced fairly high and not worth buying and maintaining unless you are dealing with a lot of the same breakers.
There are plenty of arguments on both sides about whether or not secondary injection testing is truly testing the breaker. I would have to agree that some testers perform a more realistic test than others. And you aren't quite testing "everything" but with breakers where you can see the current readings on a display, we can instantly see whether or not the CT's are working once the breaker is back in service anyways. In the case of draw out breakers where we can crank the entire breaker assembly out and we need to do that to service them anyways, primary injection makes a lot of sense. But in the case of bolted in breakers which is what MCCB's are in almost all cases, actually physically removing and reinstalling them for testing is very time consuming and entails risks of damage to the breaker or the fasteners, and takes a lot more time than draw out equipment. So in many cases secondary injection testing is good enough.
Another test, and this one does not need an outage to do, is infraref inspection and for medium voltage, corona testing. Infrared inspection is the easiest way to pick up on loose connections in equipment among other things. It is inexpensive especially compared to what the repair costs are if you ever have an arcing fault blow equipment apart. Corona inspection is the twin sister of infrared. While infrared picks up on problems with loose connections, corona testing looks for corona or minute almost invisible arcing called partial discharges that happen when insulation starts to fail. Corona testing only works above about 2000-3000 V though and it sounded like your focus is on 600 V or less, where corona testing does not work (not enough pressure).
