While there isn't a "one size fits all" answer to this, there were standard procedures followed for the delivery of parts and acceptance of those parts (and vehicles) at the depots prior to their shipment out to the units in the field. There were also standard procedures for handling of spare parts and painting of vehicles in the field.
Parts that were delivered from one manufacturer to another as part of the overall assembly process were delivered in red oxide primer (exceptions were gun barrels, those were delivered in a grey heat-resistant lacquer/primer). It's not commonly known that most German vehicles for example would have the hulls produced by one manufacturer, the turrets by another, and the guns by a third for example...with final assembly taking place with a potential fourth manufacturer if not one of the other three involved. Upon final assembly, the vehicles would be painted according to the standing orders of the day for vehicle schemes prior to delivery to the depots and acceptance by the Waffenamt inspectors. Spare parts were typically shipped to the field depots in primer only and would have been painted by the crews in the field to match their vehicle when they were installed depending on the part in question (i.e. spare road wheel).
Once the vehicles were issued to the field, it was up to the field units to maintain their vehicles. Paint and paint equipment (spray guns) were issued in quantity and vehicles were typically maintained by the crews for minor touch ups and application of camo schemes over the dunkelgelb basecoat before the introduction of three tone factory-applied schemes in late '44.
Any time a vehicle was sent back to the depots or to the factories for repair or rebuild, the vehicle would be refurbished (depending on circumstances), brought up to standard to the extent possible for the vehicle in question, repainted and re-issued to the depots to once again be sent out to the units in the field for service.
That's a high level view of the process. As with anything German related in WW2, you can never say "never" or "always" as exceptions abound on a case-by-case basis. HTH!