I debated with myself about answering. Then I decided that perhaps non-Americans or even Americans who do not understand our history will learn something valuable about our heritage.
I think I'm not a Mayflower descendant. One of my ancestors was named Warren, but I'm not sure he was related to the Mayflower Warren, or even to Pocahontas. (That's a bit of a joke, for those who follow the news.)
My earliest American ancestors date back to about 1635. They were all Puritans who came over on the Winthrop fleet during the Great Migration of 1630-1640. They settled Massachusetts, Connecticut, and Long Island, and later Vermont. They had enormous families. So I suppose there are many of us who are their descendants. I doubt most people are aware of how many Americans are cousins. For example, I discovered recently that I am a distant relation of Ron Paul and Rand Paul. We have some common Puritan ancestry going back to New England.
There is some interesting mathematics in this subject. A simple exercise reveals that if we go back enough generations, it becomes mathematically certain that many of our ancestors who show up in the family tree at separate nodes in the tree were identical. Otherwise, we would have had more distinct ancestors than the population of the Earth at the time.
Perhaps most people of my heritage who came over to America as settlers have "royal blood." That is because we are descended from fourth daughters of third sons and so on going back to Alfred the Great or William the Conqueror. What happened is that if your ancestors did not qualify for their place in the royal palace, and actually had to earn a living somehow, some of the men became wool merchants or Anglican clergymen, and some of the women married such men.
One problem with clergymen in 1630 was that if you were a Puritan, it was hard to find a job under Charles I, and you might even end up in prison, so you packed up your bags and moved your family to America. You could start a new church and be a Puritan there without getting into trouble.
One of my English ancestors was recruited by Winthrop to fortify Long Island. Before that he made his reputation as a military engineer, serving the Dutch army which was fighting the Spanish. He was involved in the very first Indian war fought by English against the natives. His family were actually friendly with the Indians in their area, and his son became perhaps the first Englishman to learn Algonquin. The natives were fighting each other, so some of them welcomed Europeans who would fight on their side.
The rest of my family is German. They came a bit later than the English. Unfortunately their part of Germany was ravaged by religious oppression, war, the plague, and poverty. In some cases their farms, workshops, and even their entire town had been destroyed by the French during the Thirty Years War. If you want to know more about the Thirty Years War, see the movie The Last Valley with Michael Caine and Omar Sharif. As bad as it seems in the movie, the reality was far worse.
There was a shorter war, not long after the end of the Thirty Years War, which began after enough orphan boys had grown up and could replace their deceased fathers in someone's army. That war, known as the War of the Grand Alliance, seems to have been the last straw for some of my ancestors from Germany, who settled in Pennsylvania. In one case that I know of, the French army had destroyed what was left of their village, leaving only the church and the town hall. The Germans had to flee and live in the woods until the French army departed.
Perhaps this makes it clear why so many Germans who lived near the French border decided to move from such a beautiful region to wild Pennsylvania. You can't eat scenery, and if you did have a farm, or a good livelihood from your craft, perhaps the French, or even other Germans, might rob you or destroy everything. Also, people were sick of religious wars, witch burnings, and torture, so they were looking for something different. William Penn promised religious freedom, along with land or employment as a craftsman. German farmers and craftsmen were in great demand. Penn sent agents to Germany to recruit them.
There were also a few ancestors from Switzerland. They fled Switzerland to Germany because at that time the Germans were not persecuting Mennonites as they were doing in Switzerland. My Mennonite ancestors were expelled from canton Bern by the Calvinists and threated with perhaps fatal consequences if they returned. I regard these people as Germans by choice, rather than as Swiss. One of them is buried today in Greisheim, a town in Hesse. I'm descended from his son, who emigrated to Pennsylvania.
During WWI some of my German ancestors changed their surname to sound more American, or at least more Dutch. They wanted to be accepted as 100% American. There was so much anti-German hysteria whipped up during that period by pro-British propagandists that in some school districts it was forbidden to teach German -- even though it was the main second language at that time for science students. Fortunately, I have studied German, so even though I missed out on speaking German as my native language within my family, I know it anyway. I could visit some of my distant Mennonite cousins in Pennsylvania and speak German with them, although they speak Pennsylvania Dutch, and I learned Hochdeutsch.
One could say that in those times the situation was suboptimal for many people. The same is true today. Otherwise, perhaps people would not move around so much. Who wants to move if your life is great where you are?
I admit to some envy when I meet people like a German girl I chatted with not too long ago. She grew up in the same house that had been in her family for several generations. She still lives there today.