Came across a town in MA with a term limit of sorts for Finance/Advisory committee, it is referred to as being put on the retired list and yup, it involves time served and age.

Adams, MA by-law on Finance Committee; There shall be a Finance Committee consisting of 15 members who shall reside in the Town and shall be registered voters therein. Of this Committee no person holding an elective position or any other position in Town, other than that of Town Meeting member, shall be eligible to serve on the Finance Committee. The Town Moderator shall appoint all members of the Finance Committee, each of whom shall serve a term of three years. It is recommended that at least two members from each precinct be appointed by the Town Moderator, to ensure equitable representation. The term of office of each such member will expire on May 1 unless the annual business meeting of the Town is not held and dissolved by May 1, in which case such appointments to the Finance Committee shall not be made until such annual business meeting has been held and dissolved.

B. 

When any member who has served on the Finance Committee for not less than three consecutive terms shall reach 70 years of age, the Town Moderator, at the completion of such member's term, may notify him that thereafter he shall be on the retired list of Finance Committee members. A member so retired shall attend Finance Committee meetings only when so requested by its Chairman, to give his advice or suggestions on matters pending before the Committee. The members so retired shall have no vote at such meetings as they may attend. Every such retired member shall have the status of and be referred to as an “honorary member.” The honorary members are exclusive of the 15 members permitted by law.

Vacancies - In the event of a vacancy on the Finance Committee, the Town Moderator shall forthwith make an appointment to fill the vacancy for the unexpired term.