Do you think you can buy 5k lots on the e-mini sp in 5-10 minutes?
You could buy 5K lots in ES in ten seconds if you are best bid and SPY is ticking down. Just about all of the big traders ICEBERG orders to fill size which is the smart way to go about it.
If stocks on the NYSE are ticking down a 5K lot in the ES is not going to stop that. All of the popular futures trading platforms that are used by big traders and firms have ICEBERG order functionality for good reason. TT, CQG, IB...
Even on the floor in the pit twenty years ago the floor brokers would scale buy or sell for the big firms and funds. And they would also do big blocks with each other and big locals for that matter.
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We could exchange some experience in doing so 

