Primer for old oil-based paint
You can overcoat over any old paint if these things are true: 


The paint is fully cured (which can take up to a year or two for single-can paints which rely on air or atmospheric moisture to cure.)   Curing is not the same as drying.  Curing is the paint molecules "holding hands", polymerizing into much longer molecules. 
You scuff-sand the old paint so the surface has microscopic ridges and valleys ("tooth"), a Scotchbrite pad will suffice, this will remove the gloss from the paint.  Be careful not to blow through corners and ridges.
You clean the surface thoroughly to remove grease, oil, dirt and the like.  Especially important in kitchens! 


Classically one might use TSP (Tri-Sodium Phospate) to attempt to both clean greases and etch the surface.

Where does primer enter the picture?  


You want a primer over bare material.  Original oil Kilz is high-performing over wood.  It's not the very best; that's a marine 2-part epoxy primer.  Over aluminum or steel you'd want a zinc chromate primer (the classic green color of unpainted aircraft), over steel I just tend to go Rustoleum.  
When you have dissimilar surfaces or colors, and you want to normalize them so they don't "print through". 
If your topcoat doesn't want to stick to the surface as-is, this is called a barrier coat.


If you don't have any of those situations, a primer may not be needed. 

I am no fan of the Kilz latex primer; I feel the oil-based product is far more suitable for wood.  Perhaps the Kilz latex makes sense over drywall; you certainly do need a primer for that.