I think that a flexible and well-designed set of rules could minimize the discontinuity; I also think that what appeared on paper to be a different character, might in practice play the same.
Powers can be broad and inclusive: let Clerics with the Fire Domain just cast any spell with the [Fire] descriptor at will; the most important currency - the number of available actions - assumes greater and greater importance at higher levels. Whether you can cast 9 3rd-level spells or 11 in a day is irrelevant; it is no longer a balance issue. Tracking it is annoying.
Assume all characters get an SR of CR+10; give all fighters Devastating Critical; have a mechanism which incorporates ubiquitous +5 inherent bonuses, blanket +6 enhancement bonuses and blanket +5 resistance bonuses - tracking this stuff is annoying. Just let everyone have all of it (to a certain point), and use this as the basis of the rebalance.
Let a new set of divergences between characters begin from this point. It can be balanced and controlled.
Okay, this sounds good for characters who've got a single concept, or have reached the peak of their Spellcasting: archmages, head clerics, rogues/assassins, warriors.
What about mixed-concepts and multi-classed characters? For example, I have an eldritch knight I'm fond of.
Another example: A character with only a few levels of Cleric and the rest are, say, Monk.
In theory you could have a number of traits and a menu for things the character qualifies for (ie: combat, magic, divine, skills, druids, bards, rage, summoner, etc.). Then they have a limited number of things they can choose, but it's stuff like access to Fire domain spells at will, or healing at-will, etc.
The problem is it's instant mastery where it only lip-service was paid before. That's not bad, if the Epic setting is exaggerated in all things (ie: anyone who's a Rogue could potentially have all sorts of rogue traits; and they all operated at Epic bonuses/ranks).
How far have you gotten with this idea? And is it Pathfinder-based? (feel free to PM me)
Ultimately, however, this is a different setting with totally new conventions and scale. In theory you could do the same by just re-skinning every Bestiary monster as an "epic" monster, and all the equipment as "epic magical hemp rope"; that's what you'd be doing, but with bigger numbers.
And lots of automatic successes (ie: you auto-kill all non-epic creatures in 1d4 hits max; or you auto-stealth or auto-climb the walls).