How big percentage is the "illegal" mass of the travellers?
When there is no border control, there are no illegal travelers, just legal.
Are they wanted beforehand or do they commit crimes where they see fit?
They commit crimes when necessary.
The people who commit crimes and bring "diversity" are allowed to the (at least in Europe) countries legally. Yeah, I mean the "culturally enrichening" folk.
Yes. Millions of such people have entered to Europe since 2015 through Europe's outer controlled borders, which were not controlled. So Europe broke it's own legislation and sovereignty too.
There are criminals always and everywhere, some are just born with a brain like that. Some don't have empathy and there are always people who want to get by easier.
That is why more criminals are not needed to enter into any country and the border control is the path of the least resistance.
Closing/opening borders doesn't change much in that.
The border control is not a "closed/opened border". It is the border control. Anyone with proper documents are able to go through border.
The world is getting more and more digital every day, soon a person wanted by Iraq will be wanted by Latvia as well, with exchanging information. Oh wait, we do have interpol already?
God bless we have.