I cant say anything about any brand except for Mobil 1, and the M1 HM oils are pretty much as MGreigor has posted.
Compared to their standard 10W-30, the M1 10W-30 HM is actually a heavy duty oil. It has a higher cSt viscosity at both 40C and 100C. It has an HTHS of 3.66 compared the regular 10W-30 HTHS of 3.14. It has 1000 PPM of ZDDP compared to the standard 800 PPM. It has extra antioxidents and extra detergents. It also has extra seal conditioner.
I did a web search(yahoo) on seal conditioners as I took over the service of my Priest's car which had 73,000 at the time and had had only a 5w30 mineral oil that I did not like, since new. You could stick your finger inside the oil fill cap and there was a lot of deposits. Basically, the results of what I learned from that search is that group IV synthetic oils tend to clean the residue of mineral oils from seals, penetrate the seals, and seep. Evidently this was a problem for folk that switched from mineral oils to synthetic oils many years ago when synthetics became widely available. To compensate for this, synthetic oils have seal conditioners to keep the seals pliable.
If you start out with a group IV synthetic(I simply know nothing about group III) your seals will have been getting seal conditioners from day one. If you switch to a synthetic after thousands of miles of mineral oil, the synthetic will clean the mineral oil residue away from the seals, which has been preventing leaks, penetrate the seals which are actually not as pliable as new, and seep. To compensate for this, synthetic HM oils have even more seal conditioner.
I sumply do not know about switching from a standard mineral oil to a high mileage mineral oil.