score:5
Yes, you can, it even seems to be the expected behaviour.
All three alliances have restrictions and rules, from basic expected ones like a maximum number of miles to some random and precise ones like a forbidden combination of carriers ("When a QF (Qantas) marketed, NU (Japan Transocean Air) operated flight is part of an itinerary, IB (Iberia) ticket stock cannot be utilized.").
Overall, the same basic conditions regarding flight planning apply to all three alliances: you should get the first ticket about a week before your first flight (a priori a firm booking with a specific flight on a specific date), you should plan all the stops you are going to make, in advance, but not necessarily the dates (and exact flight number, or even carrier).
It does not mention precisely how to make these changes for free (online or at the airport), even though the mention of "local fees" probably means you can at least show up at the airport and book a specific flight for an extra fee.
Here you can see in details each alliance policy and conditions page:
So in the end, the change of flight is a basic allowed modification of round-the-world tickets. However, be careful with all rules that apply, it might be worth reading the terms and conditions when you picked an alliance, as buying such a ticket involves thousands of dollars and may impact months of travel.