"The Pilgrim fathers commenced the colonisation of North America in a big way, but by no means did any sizeable porportion of the population of Europe leave. They mostly went on living their regular lives. Only a few brave or crazy sould crossed the pond. The population of Europe did not reduce because of migration, it continued to grow.

Culturally, the situation with space is likely to be similar, especially given the energy cost of putting ONE person into space."

First of all, my normal assumption is that Earth population will continue to grow.
But my point of my post was that *assuming* it was desirable to have a low human population on Earth, then having a space frontier is about easiest and quickest way of getting to that point with capability maintaining a low Earth population for thousands or millions of years.

As for the history with Europe showing future population trends and migrations. I don't think Europe ever desired to have a low population- as there are consequences of having a low population- such lack of military and economic power for the State. And the declining European population at present is to some degree a driver in encouraging immigration to Europe.
Second because of technological advancement- getting to space would be easier than a month long boat trip of a few centuries ago.

"The amount of fuel that must be burned to get one person into space is about the same as that person would use in a whole lifetime on Earth."

If you used 500 gallons per year and for 20 years that is 10,000 gallons or about 60,000 lb of fuel.

As I've said at various times in the past, rockets are hideously inefficient in terms of fuel cost, and to lift the fuel and gain the velocity of orbit, it requires about 9 lbs of fuel to lift 1 lb. Which means that for a human payload with chemical rockets you will never get at or below costing 9 lbs of fuel per pound of human. In other words if structure weighs 20,000 lbs you will require somewhere around 180,000 lbs of fuel, so if it's one passenger you will have a fuel cost of 180,000 lb, but if you could somehow fit in 2 passengers, you get a 90,000 lb of fuel per passenger. So, this is the general principal of why airliners want big airplanes which can hold a lot of passengers. It is also why some people want bigger rockets.

What I'm going do next is demonstrate the possibilities of lowering the fuel cost per passenger. I am doing this to allow someone to get a ballpark figure AND not as a *THE SOLUTION* to getting into space. I say this because fuel cost are not a significant factor in lowering the cost to get into space, e.g. 180,000 lbs of fuel is not very expensive- 20,000 gallon at $3 per gallon is $60,000- cheap!

So I'm going use a Saturn V [1960's technology- not that we have improved that much since:(].

A Saturn V total mass is: 3,038,500 kg (6,698,700 lb).
Fuel of first stage is about 4.75 million lbs [Kerosene and Lox].
http://www.astronautix.com/lvs/saturnv.htm
Fuel of second stage: .995 million lbs [Kerosene and Lox].
So total 5.75 million lbs.
This lifts 260,000 lb to LEO. And I'm going make most of this payload weigh in humans. In other words I am going load up passengers into a tin can and they will get a hammock and 15 mins of breathable air and nothing else. They will be crammed in, and in about 10 mins will be in space and be transferred to better quarters.
So their "tin can" could made so it's about 10% of the mass
so that leaves 234,000 lbs for human meat, some air, and a hammock. This allows 1170 passengers with average weight of 200 lbs. Which gives a fuel cost of 4,915 lbs or about 800 gallons of fuel per passenger. This is about same amount fuel used by a car which has 20 MPG and travels a distance of 16,000 miles.

Now, this could actually be made much more efficient and still be using chemical rockets. And of course it's unlikely anyone would carry 1000+ passengers and to be crammed in like cattle- assuming the people had any choice and weren't living under some totalitarian regime {like the UN or something}.
So, with better efficiency, and say passenger size of couple hundred with some creature comforts and with total flight time from the loading gate to the departure gate of about an hour. You could probably still get close 800 gallons of fuel per passengers. But then again. You probably won't even actually see this, anymore likely than that you will see people crossing the Atlantic in coal driven passenger ships. More likely with chemical rockets you get up to about 20-40 passengers and with higher fuel usage of somewhere around 2000 gallons per passenger. With higher volume of traffic that could make 200+ passenger rockets economical [you have the traffic that justifies it], it's likely that other types of technology would be used.
[And no, I won't include things like pipelaunchers, because I talked enough about them recently:). I suppose I could mention space elevators- but I have my doubts about them.]

Now the energy needed to get into orbit is only equal to a few gallons per passenger, so in terms of saving fuel, something like space elevator could save a lot energy. And by "something like a space elevator" I mean any kind of structure the stands more than 100 miles high. But as I said earlier, saving money on fuel cost is not important if the costs [to LEO] are in the $1000+ per lb range, and even at $100 lb per range it's not critically important. If it gets down below $100 lb, and one *would have to be* dealing with a high volume traffic at that point, then it becomes important/competitive advantage].

So generally this is how I see things, I would say that "opening the space frontier" is dealing with launch price to LEO of $2000 per lb or less.
Such things as "regular" Space tourism [going to the Moon or Mars] may be at around $500-1000 per lb.
A space boom is around $200 per lb.
And when the cost of things in space is close to being the same price on Earth, it would be $100 or less per lb. And this also around the point that order to go to a lower cost to orbit, you might need to use something other than chemical rockets. Any significant amount migration into space would only start after launch cost was at or below $100 per lb [one way ticket to LEO for someone weighing less than 200 lb is $20,000 or less].

So an early Pilgrim/emigrate to somewhere is space may be using around the same amount fuel as about 2000 gallons used by typical American consumer in a few year of driving in order to just get to LEO. Whereas the millionth emigrate to space may not be using chemical rockets. And certainly the billionth emigrate would not be using a chemical rockets- and that amount of energy needed to get to LEO could easily be less energy than 50 gallons of gasoline.

"A few tens or hundreds of space launches every year may be insignificant in their effects on the upper atmosphere, but tens to hundreds of MILLIONS of launches?"

I would be more worried about slowing down earth's rotation:)
Well, tens to hundreds of millions of launches per year is quite a lot. That might be one of best reasons I can think of for a space elevator.
Now I admit I haven't really given the idea of moving billions of people off earth, nor much thought to this potential of such a high launch rate.
So right now we at the tens of launches per year. By the metric I used above:
Opening the space frontier would be around 50-100 launches per year.
Space tourist 1000-1500 per lb range could around +100 space launches to orbit and maybe 100 or so suborbital launches.
The space boom at $200 per lb maybe around 1000-2000 launches per year and plus maybe more than 1000 suborbital flight per year.
So, I would say anything over 10,000 space launches per year would classify as post boom phase.
If you have 10,000 launches with 30-100 passenger per year you are moving 300,000 to 1 million people per year, so that would not be enough people to make much of a dent.
Now, last year in the world there were 2.2 billion passengers traveling on airlines:
here
You wouldn't need that many people going to live in space per year. But probably need about 100-200 million leaving Earth per year and overall traffic per year involving destination to and from space could around 1 billion passengers per year.
So if instead you had the average "super saver" emigrate spaceship holding average of 500 passengers you could adding 400,000 flights per year.
It might a problem, and perhaps a space elevator could move a lot of passenger, but the good news is you would probably reduce air travel and air traffic on Earth.

"the only way to get population under control on Earth is to balance the number of births and the number of deaths.

There are only two ways to do thius:- have fewer babies (The modern Chinese solution), or let/cause more people to die - the historical solution.

This problem is essentially solved in the first world through education of women, high quality of life, and contraception. In the third world, we face a problem that will dominate the future of the whole globe."

I think my way would be faster and cheaper.
Say an added govt cost of 200 billion* and get it done in less than 50 years- assuming of course this was thought to be a good thing.
* I'm assuming this done somewhat wisely and during those 50 years you would continue NASA spending at about $16 billion per year [btw, that yearly budget cost would add $800 billion during that time].