Mitchell has made the gesture as a “thank you” to fans following the “tremendous backing” given to Eddie Howe’s team at Doncaster on Saturday.
Cherries will make a return trip to Brunton Park on Saturday, December 1 having won 4-2 at the venue at the end of last month in a league game.
Non-season ticket holders shelled out £35 to make the 680-mile round trip to Cumbria 18 days ago, while season tickets holders paid £28. The away attendance was 156.
Subject to demand, Mitchell also said the club may offer a light breakfast to supporters before the coaches depart Goldsands Stadium at the crack of dawn.
Mitchell told the Daily Echo: “I thought the supporters gave the team tremendous backing at Doncaster. Our fans were very vocal and it made a difference. I am sure it helped the players and helped us win.
“As a thank you and to show our appreciation, we would like to offer free coach travel to Carlisle in the FA Cup. We have got three long away trips one after another and I hope it will help.”
Cherries’ trip to Brunton Park is sandwiched between long-haul journeys to Bury on November 24 and Scunthorpe on December 8.
Meanwhile, the club today announced plans to hold a series of discussion groups as part of its commitment to supporter liaison, with the first topic being the matchday experience.
General manager and supporter liaison officer Liz Finney told the club website: “This is a fairly meaty topic to start off with as it encompasses many important touch points such as ticketing, catering and value.
“But we genuinely want to provide the best possible matchday to our existing supporters and give ourselves every opportunity to attract new supporters to the club.
“Our supporters are the best source of information to help us do this effectively.
“The key throughout will be to be informal but focused, realistic on what we can achieve now, but ambitious for what we want from the future.
“We are also planning to hold a discussion group on communication and one on our commercial operation so, if these pique your interest, you might want to hold out to be part of those meetings as the same supporters will not repeat across groups. This is an opportunity to be involved so please join us.”
Fans can download a form by visiting www.afcb.co.uk/fans/supporter-liaison/