Moyes' men will be the first team to face new boss Mauricio Pochettino on Monday, following the sudden sacking of Adkins who achieved back-to-back promotions and appeared to have turned things around for the Saints after a shaky start to their Premier League campaign.
Adkins' departure, followed by the imminent takeover of former Espanyol boss Pochettino, has shocked fans and players alike, but Moyes does not think it will make his side's task any easier when they travel to St Mary's on Monday.
"It's a really difficult one to judge because I actually think Southampton have been playing quite well. I'm not quite sure what effect it will have on the players but when you get a new manager you get on with it," he said.
"I don't think players mope around too long about the manager, and I mean that with real respect for Nigel, but I think players just get on with their job. I would expect them to do that and it will be difficult.
"I don't know what the atmosphere will be like, I would have expected the supporters to have probably been in Nigel's camp because of the work he'd done there."
Pochettino kept Espanyol in the Primera Division for four seasons but was sacked after a poor start to this term, and Moyes believes his knowledge of Spanish football may have been a factor in his appointment.
The Scotsman said: "I've seen his teams play. I think Espanyol had periods where they did very well considering what was available to them.
"I don't know him personally but he looks as if he's a modern, young coach who will have a lot of knowledge of Spanish football, which will help him.
"You can see it's helped Swansea with Michael Laudrup coming in, it's given them an opportunity to have a little bit of a headstart on players at the lower level who could possibly step up and I would expect him to be very much the same."
Southampton v Everton is live on Sky Sports 1 this Monday. For more details click here http://www1.skysports.com/footballonskysports/8395413?DCMP=FootballHub_SouEve_Editorial