Chelsea missed out on the chance to go 12 points clear at the top of the Premier League table as Burnley came from behind to secure a 1-1 draw.
Burnley nick a point from league leaders Chelsea at Turf Moor.
Moyes hopes to foster a new team spirit during a four-day bonding trip to set up his squad for the run-in.
Allardyce, who steered Sunderland to Premier League safety last season after being appointed in October, has never been relegated from the division.
Everton were unable to break through against Middlesbrough, with goalkeeper Joel Robles their best performer on an uninspiring day.
Zlatan Ibrahimovic wants to lift more trophies before his first season at Manchester United is out.