Baba and Amir leave Afghanistan for the US via Pakistan because of the Soviet invasion. Baba couldn't stomach the idea of his country being ruined physically and ideologially by the Communists; so he leaves before it is too late. As a capitalistic businessman, his life and Amir's would have been in danger.
In answer to your second question, their migration is both. Baba chooses to leave before he does not have a chance to leave, but he never would have left his homeland if it had not been for the Soviet invasion and destruction.
I would argue Baba and Amir were forced to move out of Afghanistan because of the prevailing dangerous conditions. The Afghan community was divided and civil liberties were restricted. People who talked about the situation disappeared. The area was in an active state of war, with civilians suffering the brunt of the engagement. This made Afghanistan unsafe for its residents.
Baba and Amir left their home without informing their servant. They carried only two suitcases and left everything behind. The complexities of traveling at the time forced them to seek the assistance of Karim, who was in the business of smuggling people out of Kabul. Military checkpoints were mounted strategically along the different routes, manned by both Russian and Afghan soldiers. The smugglers had to bribe their way through such checkpoints. Baba and Amir arrived in Pakistan and later made it to the United States.
Baba is a big business man in Afghanistan, so he knows with the Russians beginning to invade, he has to make a choice whether to stay or leave. Baba hates to leave Afghanistan. His whole life and business is there. He friends are there, yet he knows he has to keep himself and Amir safe. He decides to go to Pakistan then onto the United States. It is not a forced move, but it isn't entirely voluntarily either. Baba knows that when the Soviets move in, then he would be force to leave, or an even worse outcome. It is a sad and difficult decision he has to make.
Amir is alright with leaving. He feels he is leaving behind the terrible secret he has kept. He doesn't realize that this secret will haunt him for many years. This was a hard time for many locals. They had to make a choice, and only the ones that were wealthy, really had the choice. The poor and uneducated didn't have many choices. Baba hates to leave Hassan, but he feels that he has to keep Amir safe.
What is tragic about all of this, is that Baba is leaving behind another son, one that has had a horrible tragedy happen to him. Instead of the being honest about Hassan, Baba chooses to ignore his other son, and only save Amir. What a horrible choice this must be for many people in these circumstances. If Baba had taken Hassan with him, would things have turned out differently? One will never know, but we can all guess the answer to this question.