"Something ends, something begins."
Translation:Coś się kończy, coś się zaczyna.
Usually both versions work, although sometimes one will sound better to a native speaker and it's hard to explain why.
Here it surely should be before the verb. Why? This could easily be divided into two sentences. And neither "Coś kończy się" nor "Coś zaczyna się" sound really well, you shouldn't put 'się' at the end of the sentence if only there's another (correct) place to put it.
Moreover, as those clauses differ just by one word, it makes sense to put it last to stress the contrast between the clauses.