Now the search is on for the Blues to find their ninth permanent manager since majority owner Moshiri arrived at the club in February 2016.
Allardyce was one of the previous incumbents of the Moshiri era. Taking charge in November 2017, he guided Everton to an eighth-place finish before being axed at the end of the campaign.
More than £500 million in transfers has been spent by Moshiri yet the club stares relegation in the face.
Speaking to talkSPORT, Allardyce believes that it’s not just Moshiri or Lampard who is responsible for the current plight - but everyone at Goodison.
What’s been said
He said: “I think that it's direction has been up and down like a yo-yo for years now and because of the lack of success and the success that the money, unfortunately, hasn't brought, because there has been a lot of money spent, they keep moving managers on and results haven't been good enough.
“The only opportunity to be successful is to change manager. I sometimes feel that that doesn't get you anywhere, in the end, other than the position they are in at the minute. Money is much tighter than it was before, financial fair play, and the manager has come up against a bit of a barrier to try to make Everton the team the fans demand and want. Of course, the manager comes under pressure they've moved him on.
“I really enjoyed my time but it was tough and has been tough for every other manager since at some stage. What they do and who they choose now is the most important thing to get them out of trouble this year. That's all it is, building for next season can only happen if someone gets them out of trouble this year.
“The problem is the lack of goals and, of course, a very leaky defence at the moment. You've got both ends of the pitch as a manager or coach you'd have to sort out straight away.
“It's a tough old world out there for finding the quality of a player in the Premier League. It's a worldwide search now, you can't just go like the old days into the British Isles and Europe, you have to go worldwide to find your players. Most Premier League teams are made up of foreign imports.You have to have that worldwide organisation to find the right players.
“When you are desperate and panicking to get players in is when you make the most mistakes. Everton have been in a panic for the past two or three seasons.
“Farhad Moshiri has spent the money. You can look at that and say they should have spent it better but he's not responsible for spending the money is he. He's responsible for providing it, not spending it. The infrastructure in terms of recruitment has changed as much as us managers have been changed.
“Stability and good organisation in any business or football club is the answer to success. Everybody is to blame at the moment. It's not just the coach or the players. Everyone at the club needs to pull together a lot better. Is everyone pulling in the right direction? You know what it's like when the Chinese whispers start going around and it doesn't help the club get where it wants to get.”