Come + gerund is like other verbs of movement followed by the gerund. It means that the subject is doing something as they move.
- She came running across the field.
- The horse came racing around the corner.
- My dog always comes bounding out of the car.
- When I call you, please come running.
Come + infinitive can mean that something happens or develops, perhaps outside the subject's control.
- At first I thought he was crazy, but I've come to appreciate his sense of humour.
- How did you come to be outside the wrong house?
- The book has come to mean something quite different for me.
- I have come to see that you were right all along.
Come + infinitive can also mean that the infinitive verb indicates the purpose of coming. This is a similar usage to other verbs of movement followed by the infinitive.
- I came to see the new exhibition.
- We came to watch a movie.
- Will you come to eat dinner?