Hatters manager John Still informed the pair of his decision on Wednesday and replacements are due to be installed in due course.
Neilson, who appeared 63 times for the Hatters as a player, originally joined the club in a coaching capacity as youth development coach in the summer of 2008.
The former Wales international defender performed a number of key roles, including three spells as caretaker manager.
Meanwhile, ex-Chelsea and Russia international Kharine has been on the Town coaching staff for almost 10 years, and helped aid the development of all of the club’s goalkeepers – both senior professionals and youth team keepers – during that time.
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Still told the club’s official website: “Alan and Dmitri are two fantastic men
“Both have been great servants to this club.
“In my time here so far they have shown terrific support and have proven to be honest and trustworthy individuals who I really consider to be friends of mine.
“However, when I took the job, I was promised that I could bring in my own staff if I felt it was needed. Now the time has come where I am now able to bring these people to Luton Town.
“I’d like to wish them both well in their pursuit of new roles – they will certainty leave here with a glowing reference.”
Town managing director Gary Sweet added: “It is difficult to put into words how grateful we are to both Dmitri and Alan for their loyalty and professionalism whilst they have been with us. They have almost become part of the furniture.
“They both are authentic, hard-working, and respectable people who have not only acted professionally during their term at Luton.
“It’s really tough to see them both leave us, but difficult decisions have to be made in football. John Still is the manager of Luton Town and we trust and back any decision that John makes if he feels it is for the good of the club we are all committed to.
“On behalf of all the staff at Kenilworth Road I would like to wish both Alan and Dmitri all the very best of luck in their careers – not that they need it given their credentials.”
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