How to transfer data from/to the I2S controller through AXI DMA
Okay, it seems my understanding described in the question is correct. The AXI DMA does work as I described and the I2S controller also seems to work correctly. The problem was that I thought that once TLAST is asserted it resets itself and the transfer continues, when in fact it immediately stops. I should've read how AXI stream works. 

So I needed to set the period register in the I2S controller higher (it is only 16 bits long, so I might make it bigger later). Now I have another problem with MM2S transfer, but that doesn't really belong to this answer.