What might have been some of the reasons people began moving around the world

The main reason for migration is mostly food: people migrate where they can produce food easily, catch it, or in modern times, to produce money to buy food.

Other reasons is conflict with other people and competition for food: people migrate in the hope that elsewhere they won't have to compete.

Finally, people might also migrate to look for a better climate (where they don't have to built very cold resistible houses for example).