The Warriors long-serving scrum half will join up with Bristol in the summer after signing a two-year deal.
Warriors' coach Sean Lineen paid tribute to the six-times capped half back and said: "Graeme has been a great servant to Glasgow and to Scottish rugby. He has been invaluable to us this season with his knowledge and leadership skills. I know the boys will be sad not to hear his chat next season.
"I've known Graeme for a long time, back to my days at Boroughmuir. My last season as a player was his first season – he talked a lot then too. I'll be sorry to see him go after this week's news that we will be allocated extra funding to strengthen the Glasgow Warriors squad, but it's a great opportunity for Graeme at this stage in his career", he added.