So far I have not seen any site or exchange offering this. We know everything in cryptocurrency is about money and its business. I think there might be some instances for this but as to have a really free transaction, it is quite impossible. Energy is wasted in mining so it is difficult to give it for free.
You really cannot tell that there was no free transactions available earlier, most of the online wallets used to transfer funds for free without any fees before the transaction charges started increases because of the block limit and getting to a consensus took a really long time which made a lot of difference on how bitcoin is adapted since, now every online wallet does charge you fees for every transaction and nothing is free now, If you are using your own wallet then you can send transaction with very minimal fees, less than ten cents right now but without any fee is impossible if you are planning to send your coins 
.