I don't know of a thorough design guide of the sort that you're likely seeking. I have this but I've deemed it too time consuming for practical application:
Link. It is fascinating theoretically, however, in that it addresses the interesting question of how much weld do you need to prevent pure compression buckling when that buckling technically produces no weld demand until after movement is initiated. Neat. Anyhow, here's what I recall having done in the past:
1) You don't want the add on pieces buckling ahead of the composite member so you keep your kL/r on the new pieces less between welds less than the kL/r of the composite section with some margin of safety.
2) Often you need some of the axial load to be moved from the existing section into the added elements of the new composite section. This should happen as quickly as possible at both ends of the member. Lengths of continuous weld as needed to get the job done. Similar to BA's point.
3) Do the usual VQ/It business for the actual flexure that you need to design for. This will be your bulk stitch welding.
4) Address the weld demand created by compression buckling by throwing a lateral load of 2-5% of the axial load on at mid-span and then back to VQ/It for that load. It really doesn't take much to get this job done.
If your applied moments are anything at all, #3 will dwarf #4 and you can sleep easy on that.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.