I have a kettle sour going right now... kettle sours are just that, soured in the kettle. This means that on the cold side the routine is the same and there is zero problem with cross contamination. My routine is all follows.
Mash as usual. Boil for 5-10 minutes to kill any unwanted yeast or bacteria. Chill to 80 or so degrees. Pitch lactobacillus plantarum (Swansons caps or good belly probiotic shots) or your blend of choice from your LBS, keep warm for 2-3 days checking pH (test strips suck by the way). Once at your sourness of choice (3.6 is mildly tart, 3.4 is tart, 3.2 and below is sour) fire up the kettle and boil as you normally would. The rest is no different than any other beer.
Couple suggestions / FYI's.
The Swanson's L. Plantarum seems to form a pedicle in the kettle and are slow to sour.
Good belly probiotics are the same bacteria, but there is no pedicle in my experience and they sour faster (higher cell counts).
Plan ahead. You need to be able to start your boil when the wort is at right pH... if you leave it even 12 hours too long you may be more sour than desired.
Yeast seems to take off much slower in sour beers, so don't be surprised when it's not as vigorous a fermentation.
Use a simple grain bill. 50/50 white wheat and pilsner or 2 row is common.
Keep hops lower IBU.
To answer your question. The wort stays in the kettle, no need to transfer for "souring" portion. I haven't found any issues with off flavors due to oxygen, but if worried about it you can always purge with c02.