Muireadach gained the Mormaerdom by challenging the rights of the current Mormaer, his elder brother, also called Muireadhach, hence Muireadhch Mór (in English, "the elder").
The case apparently went to arbitration, and the king decided on the right of Muireadhch Óg.
On 13 December 1213, Muireadhach Mór resigned the Mormaerdom, taking lesser lands and titles in compensation.
In a document dating to 1226, Muireadach is referred to as "Sheriff of Stirling".
Muireadach died before January 1234, when his successor appears with the comital title for the first time.