Emigration is not entirely random with NRaas SP as it is with the EA version of progression, and it is entirely shut down by default. There has to be a reason, one of about four, for a household or some individual sims to emigrate. Note that once emigration happens, those sims will still have a history in your game but they themselves are out of it forever.
The three main scenarios that can run to cause emigration, if they are allowed to, are described here. One that is not described happens when an individual sim has generated a lot of enemies and a Run Them Out Of Town, also known as a Lynch Mob, scenario is running.
StoryProgression-FAQ-Immigration#How-is ... y-this-mod
To prevent emigration of particular sims, NRaas > SP > Sim (Town, Household, Caste) Options > Household: Allow Emigration > False. On sims for Sim and Household Options, City Hall for Caste and Town Options. Alternatively, you could set the Town Option to False and mark the sims, households, or castes you do not wish to protect from these scenarios with True as Sim, Household, and Caste options will always trump Town Options.
To encourage a particular household or caste to emigrate "right away," meaning during the next cycle of the relevant SP Manager so that could be one or more sim days later, set their Emigration Ratio under Household or Caste Options to -1 as it is impossible to have a debt ratio below that threshold (ratios cannot be negative).
Note that the SP Population add-on module is required to manage emigration as many (or all) of these settings require it. The Debt Scenarios I believe will require the SP Money module.