As mentioned above, your transformer should be fine unless it is overloaded. A minor overload, generating excessive heat will degrade the insulation, and potentially lead to turns shorting. A major overload will burn open one of the windings entirely.
The conductors themselves do not degrade in their ability to carry current in any capacity as long as they operate within their ratings. Again, passing 1000 amps through a piece of #30 copper isn't going to work for long, but that is because the heating will melt the conductor.
The primary failure mode of conductors, be they aluminum, copper, or whatever is an enviornmental breakdown of the insulation systems, or in the case of medium to high voltage (15KV to 345KV) insulated cables improper installation, typically not observing the minimum allowed bend radius.
Environmentally, the most common cause of insulated cable failure is exposure to ultraviolet light, with a close second of exposure to oils or other chemicals that break down the actual insulation jacket. Very rarely there can be failures due to water intrusion, but again, that isn't actually a failure of the conductor, but rather the insulation system.
To be complete, there is a possibility of actual conductor failure, but I don't consider it to be a true failure of the conductor personally, from improper installation leading to galvanic action or corrosion (think wires on an automotive battery terminal corroding from the acid, or copper developing it's typical green patina when exposed to salt water). Those failures almost always happen at a connection point, and are the result of poor or improper connections.