If you know the index rules, it's fairly simple to calculate the turnover it is going to generate when it rebalances and subsequently the impact the rebalance is going to have on prices. This is a fairly well known behavior and a lot of players have been doing this for a long time. Basically every index of any significance is already being monitored and rebalancing effects are "front-run" this way.

I put the word in quotes, because this is a perfectly legitimate way of front running. The major indices are all public, and anyone can take a crack at this. In other words, if I announce to the world a month in advance that on a specific day and a very specific time of the day (at the closing auction) I'm going to buy X amount of specific stocks and sell Y amount in other stocks, I can't blame people for using that information against me.

How do you front run a mutual fund?

as a "consumer" of the mutual fund, you can't.

but the mutual fund itself makes large trades: somebody downstream executing those trades or having access to that data could front run.

You mean an index fund?

Predict or react to index changes faster than the funds that are compelled to follow the index.

But you can't front run shares of mutual funds; they always trade at close of business at NAV.

You could potentially front run ETFs, but if you're worried about that, you can use limit orders and get the price you want or not transact. As long as you use a competent broker that offers limit orders.