Splitting Movement and Support
The key idea here is that you first resolve your march order. Then, if you march one or more of yours units "into an area containing units from another House, combat ensues" (rules, page 17).

So, you must first move all your units to the areas that you want. Then if you did move into an area that contains units from another House, you resolve combat. This means that units moved into King's Landing would be able to support the combat as if they were there in the first place.