I suspect that an equity desk takes the block, parcels it out into lots of small pieces, and works the market to get it off their books. Of course, they charge for that service, both in commissions and spread. So effectively, instead of paying HFT firms for providing liquidity, they're paying an investment bank equity desk. Does this really save money? If it does, why all the hoopla about HFT if you can avoid them by going through an equity desk?
Don't tell me that they just put their block on IEX and the tooth fairy fills it without price impact. That would be some serious magic.