Which websites have hit you with that kind of thing - where they disable your withdraws until you provide I.D?
Poloniex, I forgot exactly how long it took but it was very annoying, not only because I had to do KYC but it had to be done over and over again, it was in a pending then review suddenly back to first step which I had to provide the document so many times, open tickets support were only answered with same answers over and over again, I will leave my funds there if only a few dollars but this is quite a lot for me.
Finally after a long wait my status verified immediately withdraw all the funds right away and just trade on the local exchange and still need to do KYC. This is the policy of all CEX and cannot complain about this.