MySQL replication lagging behind with log_slave_updates

I'm in the process of setting up replication in this topology:

   /--B
A-
   \--C--D

In order to achieve this I need log_slave_updates to be enabled on C, so D will also get the changes from A.

C is a pretty beefy server. Amazon EC2-based. 32 cores, 250GB memory and 20000 IOPS data volume. It's more powerful than A, which has less CPU power, memory and 3200 IOPS on its main disk.

As soon as I turn on log_slave_updates on C, it almost immediately falls behind A. This is odd to me because I would expect it to easily outperform, and in my mind log_slave_updates is just writing its log to disk.

I don't know where to start debugging this.