Let me try to answer. I have worked at an Algo-trading firm that trades equities and have seen how trades are executed at the order book level. Let's say the price of the stock is 100 (last traded price). Let's say the order book is as follows:
Bids: Bid1 = 99 (size = 10,000), Bid2 = 98 (size = 20,000), Bid3 = 97 (size = 25,000), Bid4 = 96 (size = 30,000), Bid5 = 95 (size = 40,000): total size = 125,000 stocks.
Offers: Offer1 = 101 (size = 10,000), Offer2 = 102 (size = 20,000), Offer3 = 103 (size = 25,000), Offer4 = 104 (size = 30,000), Offer5 = 105 (size = 40,000): total size = 125,000 stocks.
So the bids and offers are symmetrical in this example and the price is in perfect equilibrium. Imagine two scenarios:
Scenario 1: an aggressive buyer comes in and puts in a buy order with a price limit of 104 for 100,000 stocks (which is more than $10 million notional). If the execution of this order is "stupid", it will instantly hit all the offers until price = 104 and suck out all the liquidity up to that price. The buyer will only fill 85,000 of his 100,000 order, the price instantly moves to 104 and most likely the offer at 105 will disappear and move to 106 or higher.
Scenario 2: same buyer, but smart execution: the buyer uses an Iceberg order (this is an order where only a partial size appears in the order book and when it's fully hit, the size keeps "reloading" until the entire buy order is filled). The iceberg order is bid at 95.5 in size 10,000. It will sit there for a while until it gets hit by an offer. After a few hits (maybe 25,000 total pieces) the offers might "freak out" and re-quote best offer at 101.5 or even 102, at that point the Iceberg disappears and comes back later when the price action has calmed down.
Scenario 2 can go one for an entire week, until the buyer is filled. Unless the buyer is unlucky and the entire market is rallying at the same time (so the price action goes against the buyer), it is likely that a very large size can be acquired over time without the price moving too much: that's what smart execution is all about: satisfying buyer or sellers, without moving the price too much.
Obviously, as explained already, when the entire market moves too much based on fundamentals (i.e. recent action), it is difficult to carry out smart execution and the execution algos have to get more aggressive, moving the price more.
In conclusion, large orders can move the price a little (difficult to say by how much): but usually large orders are carried out via smart execution and in fact the execution algorithm gets remunerated for carrying out the order in proportion to how much the algorithm moves the price whilst actively carrying out the order (so it is in the execution market-maker's interest NOT to move the price when filling an order).