People have all sorts of reasons to move great distances.
Before the development of the impeller drive and hyperspace travel, it would seem that you would have to have been fairly wealthy to be able to flee earth. Of course, there are several models that could have allowed poor or refugees to go to other worlds but many of them have less than optimal situations for said poor once they got there. Think planet sized company towns if not almost slavery since the ownership of the colony is going to be in control of most of the resources and services once you get there. Certainly there could be the kind of situations that you see written about using the American Frontier (both the expansion into the "West" and "midwest" but also initialy on the East Coast beyond the original settlements along the coast where the "frontier" was whatever was behind the next mountain range).
Look at the people of Ireland who left in the 19th Century and early 20th Century. Irleand had been effectively subjugated by the English 200 years earlier and the "native" Irish population was driven from any land holdings and were little better than surfs but without the "protections" that implied in earlier times. With the famines in the 19th Century, many of those who survived did so by getting out of the county which meant going places like the US, Australia, South Africa or anyplace they could attempt to find work & food and a place where they might be able to improve their lives.
All of those places had at least the opertunity to aquire land or be able to create a life with improvements. Crossing the Atlantic for many of the Irish (at least) in the middle of the 19th century was a dangerous proposition. Primarily because they had little if any money they had to make do with what they could find in transportation. That often meant not just sailing ships, but old ships at that, many of them having been former slave ships that had been driven out of the slave trade by the English inforcement of anti-slavery laws. If you (and the ship) made it to the US, you still had to find a job. The alternative, at home in Ireland, was a very good chance of dying since you both had no money and even if you did have "rights" to grow food in a small patch of ground, the primary food being grown for use by the Irish had become focused on the potato which gave the highest return for the small plots the working agricultural Irish were limited too. The potato blight destroyed that as a food source. While there was a lot of food crops and livestock being produced, it was primarily being done on the large estates/plantations (essentially large commercial operations with low paid Irish labor) and the food was being exported to England and elsewhere in Europe.
In that 19th Century time frame (and earlier), it was possible for poor people to get to other places even if it did cost everything they had because there was transportation available and there was work -of sorts- at the other end of the trip. When you shift that to space travel, prior to hyper-drive, It is really expensive to go to other star systems. Even with impeller drive, it is going to both take a long time and be expensive. Planets looking for new people are going to want at least skilled/trained people that fill needs in the system. This is not a situation of stuffing live bodies into ships at a rate that at least gives you a little profit on the transpiration and dumping them into some city or port to find their way thereafter.
Even those systems/people who are purchasing genetic slaves in the Honorverse are buying for speicifc needs and getting slaves "designed" to do what their buyers need.
Control for getting on an early colony ship is going to be really ridged. That would be dictated by the ownership of the Colony or -once transportation and shipping between systems becomes regular, reasonably short, and cost effective- the immigration requirements of the destination system.
