If dvt were to cause pain, does that normally present as a constant pain, or can it come and go?

It can be either. A blood clot can happen without any symptoms. But, it typically occurs in someone who has at least one risk factor-typically it occurs in someone who has immobility. The only time it happens in someone who is mobile is when there is a clotting issue and this can be found by doing blood work. Or, in someone with an underlying malignancy. These are just examples-your doc can help.