You will stay a first cousin but as it drops a generation on your cousin's side, you will be a first cousin once removed. To your cousin's grandchilden, you will still be a first cousin but twice removed and so on.
When it drops a generation on both sides i.e. your children and your first cousin's childen, those offspring will be second cousins to each other. Again if it drops a generation on one side,it wll be second cousins once removed and a further generation, second cousins twice removed.
Your grandchildren and your cousin's grandchildren will be third cousins to each other.
To sum up simply, when it drops a generation on one side only, the status remains but removed comes into play. When it goes down a generation on both sides, then it becomes the next level of cousins. In practice people tend to say 'he's a distant cousin'.