Don't really maintain mine Tony, don't seem to need much attention really ! Having said that, a squirt of dry ptfe spray on the scroll mechanism bits and maybe the rest occasionally wouldn't do any harm, and a blow off with an airline occasionally ?
As Paul says very little maintenance required, moving parts are subject to very little relative movement compared with their construction so wear is minimal.
Brush out any accumulated dust from the scroll and jaw carriers when any stiffness or your conscience dictates and a squirt of dry PTFE lub. on the moving parts. A coating of Microcrystalline wax, suitably buffed off on the outside handled areas.
Some older chucks were fitted with oiling/grease points but this only increases the dust attraction problems when used with wood.
If the chuck gets too gross a coating of sealer or finishing media a rub down with thinners and if Iron or normal steel construction and unfortunate enough to get a rust bloom then a quick spin with some 240 grit, (mind the fingers) before a re-wax.
Like above I have never needed to take apart and clean. I have a cut down paint brush just for this purpose after every use it takes seconds to brush out before putting away. Other than that a drop of thinners just to clear any finish that may have got on to it.
I have had my chuck for a good 7 or 8 years
Yes I do use other attachments on the lathe and yes I put everything away after use :lol: and the wife complains that I clean the shed more than help her clean the house :twisted: :lol: :lol: :lol: . My poor shed is not that large