The Toffees will have a new man in charge for the first time in 11 years next season after Moyes accepted the opportunity to succeed Sir Alex Ferguson at Manchester United.
Moyes is widely regarded to have done an excellent job at Goodison Park despite failing to win a trophy, and goalkeeper Howard sees no reason it cannot be a smooth transition.
Wigan boss Roberto Martinez is the favourite to take over, with Brighton manager Gus Poyet and Celtic's Neil Lennon also in the frame.
Howard told the Liverpool Post: "We are a really strong group of quality players and the foundations are set.
"The table is set for the next guy and I have no doubt, if he is a strong character, a guy who can lead us well, we will have no problems.
"Manchester United and Everton are in the same boat with having to replace the manager in terms of expectation and all the rest.
"It's just very rare to replace one long-serving manager with another long-serving one so we will need to fill the vacancy with a very strong character. And I have no doubt that will happen."
- Manchester United
- Sir Alex Ferguson
- Goodison Park