In the past, most OneWorld airlines were happy to take over an xONEx issued by any OneWorld airline, since they all publish the same fares, so one issued by any conforms to the same fare issued by any other. More recently, especially after a few super-cheap RTWs, many airlines are now reluctant to touch an RTW issued by a different airline. I believe AA will take over one issued by another airline for a fee. If you have an AA prime flight that you can upgrade with miles+copay or an SWU, then AA will take over the ticket as part of the upgrade, at which point the AA RTW desk will handle further changes without the takeover fee. Other airlines will sometimes take over another's RTW, but it may depend on which agent you get.

As noted, travel agents usually are unable to take over airline generated itineraries (but airlines can take over agency issued ones).