| Transits won't work, firstly, every country in the world (nearly) has a different mean sea level, and at the scale of a mountain, curvature and refraction claculations are not accurate enough. Curvature of the earth, and refraction, due to the different strata, densities, humidities, and temperatures of the layers of atmosphere the instrument is measuring through.
Even if measured from various points there is always a calculation erry (pyramid of error).
Gravity is different dependant on the rocks in the earths crust below the mountain ground, same goes for magnetomic - magnetism.
The world is not a sphere it is a geode, i.e. thicker round the equator, and dips in on heavy mountain ranges (e.g. the range everest is in) so measurements must only be acurate if take, from what is a truely arbitrary base height.
The answer is truely complex, and based on gravity readings taken by the US and UK governments gravity survey to calculate the corrections needed for the WGS GPS theoretical geode (the true shape of the world - ish) system.
All points of "interest" have been re-measured, since the invention of GPS by data post processing, I am not entire sure if this has been entirely redone with Real time GPS.
The earth has also of course been "measureg" with Radar survey from orbit, but they can "miss bits", so the peaks may not reflect is they do not "bounce" the signal perfectly back.
Any measure is always going to be an approximation, no matter what system. and the next time it is measure it will probably be different, as mountians are a. heavy and may sink, b. may be on or near an active fault line, so can go up or down, c. be in an active volcanic region, and again can move.
I hope this is a sufficient answer, but the answer is measured by various means, repeatedly.
A "local survey" to the countries local Mean sea level, may be different, but always within a few inches the world average, thanks to the imperial actions of the major countries in the 19th century, as they all measure what they had conquered, normaly for military reasons, but for instance the UK "Ordnance Survey" which retains the military Monica. | 
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