I've only shifted characters over to a new system, not a new genre or setting. We went from Shadowrun to Savage Worlds. Shadowrun's rules were just too convoluted for us. We were still looking up rules six-months into the weekly game. Savage Worlds keeps it simple.
To make the conversion, it was clear there was no suitable one-to-one conversion between the two games. The probability distributions were too different. The characters would have become unbalanced in the new system. We jointly decided on an experience level that reflected how much our characters had achieved to date, and simply recreated our characters from scratch, keeping the feel of the old characters.
When playing any new system, we allow respeccing any unused abilities between the first few sessions. This helps the players get comfortable with their characters. Especially when you're shifting systems, this gives players the opportunity to be confortable they're playing the same character.
There was some loss in the translation, mostly in regards to magic, which is a small heartache. After a few sessions, it's generally forgotten. Ultimately we had a better time using the new ruleset.