You are right on that it is what could be understood when saying this sentence. But until now the only accepted translation to 'springer' is 'running' in english and I will guess there should be another word for 'chasing' in swedish. Google translate gives 'att jaga' and 'att förfölja' among another couple.
What's your native language? I think they run for him is a bit strange in English. It could mean for instance that they run for his sake. They rarely use run for with people the way they do in expressions like run for the hills where the hills is the goal of the running, I think.
There is this expression, the child runs for his mother but that doesn't mean the child runs after the mother, but to her.
PS if you speak Russian, бегут за ним or бегут вслед за ним are both fine here.
That is strange. "The police runs after him" is grammatically incorrect. In American English the singular would be, "The policeman runs after him." The word, police, is a collective noun and is treated as plural - "The police run after him." We Americans are notoriously inconsistent when it comes to conjugation and collective nouns. We say "the press is" instead of "the press are", for example. In this case, it's police = plural.