I believe they can.

If you're an ESL writer, then you probably speak another language as well. Are there dialects in that language? If there are, then haven't you seen people completely change the way they speak the language so that it sounds like the dialect that is not native to them? I've seen this happen with my mother tongue. I do not think English is so different from these dialects that someone who natively speaks or writes it one way could never change and start writing or speaking it another way. In my opinion, it may even happen faster with English because luckily, English grammar rules are so well-documented. FW gave the example of British and American English language differences.
I think that if an American were to live in Britain for a prolonged period of time, then they would be able to speak more like the Brits than the Americans especially if they were trying. To me, it is the same thing if a Brit decided to live in America for a prolonged period. Furthermore, I think it is different with writing because you do not need as much time to 'sound native'. A year is enough for anyone actively trying to change their English language style to do so up to a point where their original dialect is no longer detectable.
I think the first step for any ESL person seeking to change their writing style is to recognize where the differences in language expression come from. For many, awkward and noticeably ESL phrases come from the interaction between native tongues and English. If you really want to improve your English, you should try as much as possible to think in English rather than thinking in your mother tongue then translating it to English.
In this forum, I come across these question more than I want to admit. I have even posted something similar myself. I think the answers from some ENL speakers, especially those who admit to not knowing another language are misleading. The reason I think this way is because the label ESL is something that could never change no matter how good your language command became. I mean, even if I were to become an English professor, I'd still be ESL. At that point, however, I think I would stop advertising myself as ESL and letting everyone know that English wasn't my native language. The advice from some ENL members here seems to suggest that I would still need my work proofread by native speakers even at that point. Of course that is false and misleading, not to mention too pessimistic for anyone seeking to improve.
NB. This advice is for people who have already used the language long enough to express themselves understandably. I doubt it would work for someone who is just learning English