Languages can do both and there are many well documented cases of each.
Language divergence is when people of the same language spread out and form sub communities. Over time the language develops dialects and sometimes different grammar rules over time. This can eventually, given enough time, develop new language.
Language convergence is when two different languages are intermingled in a society. Both languages influence the other and they borrow words, concepts, and grammar from each other. This is often easier to see the change occur than divergence and tends to happen much faster.
If you want any more detail or still have questions, let me know. Hope this helped.