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:
- for Star Alliance, look for "4.1. Change of Plans" in the help and it mentions that "Change of travel dates, carriers or flights is free of charge." (I suppose that "flight" means in fact flight number).
- for Skyteam, the fees for changes are: "USD 0 for date/time/flight number changes (other than the first sector) (local service fee may apply)"
- for OneWorld, "All remaining flights may be left open (i.e. without confirmed flight numbers and dates), as long as each flight is reserved prior to departure. Tickets can be booked or issued up to one hour before departure in some countries."
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.