The 37-year-old has signed a two-year contract at the Broadfield Stadium after the clubs agreed a compensation package.
A statement from Bury read: "The board of Bury FC would like to make clear that after having spent many hours over the weekend speaking to Richie in an attempt to convince him to stay, it became clear that the primary factor in Richie's decision to leave was his desire to be closer to his young family on the South Coast.
"Richie has asked us to put on record his sincere gratitude to everyone associated with the club, and especially Shakers supporters that have supported him throughout his time as our manager.
"Likewise, on behalf of everyone at Bury FC we would like to thank Richie for his time at Gigg Lane, and the huge part that he has played in seeing us gain, and establish, ourselves as a League One outfit."
Peter Shirtliff has been placed in temporary charge for this weekend's game with Middlesbrough, with Ben Futcher as his assistant.