Common Trans-axle Problems
Mostly they're pretty durable, belt breaking and wear are common problems on both walk behind and tractor mower trans-axles.
Walk behind mower trans axles are operated by pulling and holding a lever, the lever is operated by cables, and they need to be adjusted and replaced in time.
Nearly all trans-axles are oiled for life, meaning you don't need to dip or top up, if fact on most you can't, there's no access.
Hydro-static models will need an oil top up, usually they'll have a plastic oil reservoir mounted on top of the trans-axle. A special oil grade is used (20w50) to top up and air will need to be purged from the system, but it's not difficult to do.
Tractor mower trans-axle brakes will need adjustment, pads may need to be replaced and brakes not working is common.
Constant Variable Transmission (CVT) (YouTube link) are fitted to some tractors and as they age belt wear and non OEM fitted belts can cause lots of problems.