Imperator Rome migration attraction unfortunately doesn't work like this. It can only affect neighboring provinces (except by sea).

It is supposed to be slow, but not as slow as it is right now. I have yet to think of a good mechanic to get dwarves to move into the mountains.

You can get new pops in exchange for manpower with the settle down army interaction. Only works outside of the capital region though.

Ideas:

1) a mission tree for Balin that generates pops at different milestones
2) an event that fires automatically after he takes a certain number of counties in the province that either generates pops or gives Dwarven realms and event where their pops want to move to colonize Moria, and they get gold, stability, and/or other currencies in exchange, but they can also refuse (ai won't).
3) an event that gives Balin an insane pop growth for 50 years or something like that after he takes some territory (shows viability of colony)
4) a decision that targets a specific county and generates pops when it is owned