Primers (of either base) can be top coated with either latex or oil. Double check your primer label.
What doesn't work is an oil based topcoat (brittle film) over a latex topcoat (softer, flexible film).
An oil based primer is a good undercoat for exterior (or baths), due to its greater degree of waterproofing.
There are two critical steps in prepping kitchen walls.
Clean and degloss
Cleaning is critical: TSP or TSP substitute.
Deglossing: either light sanding (120 grit) or a chemical deglosser
Once a surface is deglossed, it should be primed, especially if going from one paint type to another.
A deglossed oil undercoat can be primed with a latex primer. Either oil or latex can then be used as a topcoat.
A deglossed oil undercoat can be primed with a oil primer. Either oil or latex can then be used as a topcoat.
It is better to stick within families of paint types, if you can, but going back and forth between types is possible with primers.
At this stage in coating, I'd let everything dry for a week, then sand lightly. If the previous layers are adhering, then proceed with latex primer, latex topcoat.
If,, however, they are sloughing or chipping or you can easily pull a layer off if you leave painters tape on overnight (and then pull it off), then you must sand down to the last well adhered layer and then repaint. Good luck.
Best Answer
Yes you can, but you will want to ensure the existing old oil-based paint is properly deglossed and clean to ensure a good bond with the water-based Kilz.