Everton could move into 16th in the Premier League on Monday after defeats for Leeds United and Southampton on Sunday.
A double from Jefferson Lerma put Bournemouth on the front foot early on, although Leeds were able to get back in the game before the break through a Patrick Bamford goal.
However, the hosts struck again through Dominic Solanke and Antoine Semenyo to seize control of the game in the second half, eventually winning 4-1. While Bournemouth are now well clear of trouble at the bottom, Leeds remain entrenched in the battle to stay up after a testing spell under new manager Javi Gracia.
Elsewhere, Southampton look destined to drop to the Championship after a 3-1 defeat. They took the lead at Newcastle, but goals from Callum Wilson either side of a Theo Walcott own goal gave the Magpies a big win in their push for a Champions League spot.
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The Toffees' - who are still in 19th in the Premier League - task to stay in the Premier League looked like it was going to be even tougher at one stage on Saturday, as relegation rivals Nottingham Forest were winning for long spells at Brentford.
However, two late goals from the Bees meant Forest remain in touching distance for the Blues ahead of their visit to the King Power Stadium.
Everton have been dreadful away from home in recent seasons, but will be hopeful of a repeat of last term's stirring 2-1 win at Leicester. With fixtures against Brighton and Manchester City to come, a positive result is imperative for Dyche's side.
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