There are some people who say if you can't name a replacement then you've no right to want him gone. Fair enough. That's their opinion. I'd just have the following as a criteria for being the next Everton manager (in no particular order:
1. Has to have 3 or more seasons getting the team they manage in Europa League or fighting for a Champions League place in either England, Germany or Spain.
2 Worked with young players and improved them enough so they're first team regulars or have sold them on for a fair whack.
3. Worked with limited resourses.
4. Can set up a team that can defend whether that's from the front i.e pressing or difficult to play through.
5. Able to motivate players.
6. Has a plan B and possibly C if Plan A isn't working.
Anything else would be a bonus. A manager like that should be able to at least make us competitive even if we never reach Champions League under him he'd have laid down the foundations for the next manager.
Moyes had some of the above traits hence Martinez piggybacked off his work and was successful until he decided to dismantle the defensive side in order to play attacking football.