I may be incorrect (probably not), but I believe the following action allocation was used:
Lieutenants = 1 action
Vassals = 3 actions (or more)
Vassals are domains while lieutenants are individual characters that assist domains.
Vassals have the normal actions that they would have regardless of vassalage agreements and their actions are not selected by the master domain, but by the regent of the vassal domain or the DM if the vassal is an NPC domain. A DM could allow the player controlling a master domain to indicate the requested actions of an NPC vassal domain if it would assist in domain management, but such an indication would obviously be subject to the DM's discretion. A vassal could have more than three actions in a turn as well if that domain has bonus actions for any reason, such as for having a lieutenant as described below.
Lieutenants assist a domain in the coordination and execution of actions. Regardless of the number of lieutenants a domain has beyond the first, if a domain has at least one lieutenant it receives a bonus action. This number of bonus actions does not increase for having multiple lieutenants.
I believe that multiple lieutenants can assist in actions to improve the chance of success. Each can assist in one action.
