Someone called X dit Y = X aka Y
in this case Y is a "surnom" in french , this "surnom" could be something like "le jeune", "le vieux" to distinguish the son from the father (X dit le jeune, X dit le vieux), it could be "petit" or "le petit" to distinguish the small cousin from the tall one , it could be " du pont" ( X dit "du pont" , living near the bridge) to distinguish him from another X dit "du bois" living near the wood), it could be anything with no explanation for us living in the XXI th century.
Concerning the DUPOUY dit Le Mougnon du Bas :
1. it could be Dupouy dit Le Mougnon , du bas ( living in the bottom of the village you don't mention)
2. Dupouy dit "le Mougnon du Bas "
If le Mougnon is the name of a place , I found only one in France , near Gondrin in Gers departement :
I also found a Pont Mougnon in the Deux-Sevres departement