I disagree... it is not the
proper way, but often you can get away with it. For a while. Those quickie retrofits aren't known as "Death Kits" for no reason.
The first problem is, as mentioned, that the two oils are incompatible. I don't believe they cause any clogging or foaming - but since they don't mix, and the old oil is not soluble in (nor circulated by) the R134a, it has to be occupying space somewhere in the system,
hopefully staying out of the way and just taking up room that should be filled with R134a.
The correct way is to drain the old oil from the compressor, flush all parts with a suitable solvent (R11 is often used), and of course put in a new receiver-dryer. Assuming that the compressor seals work with the new oil...
Another problem is the o-ring material, and every connection in the system has one, unless it's a really old one that uses flare sealing. I don't remember if the oil dissolves the o-rings or if the R134a attacks them.
To a lesser extent an R12 system will leak a bit faster with R134a because the molecule is smaller and will pass through the wall of the hoses. R134a is now up to nearly $10 a can. If you're going to replace hoses, get the "new" barrier (lined) hoses.
Bottom line is, if it doesn't work to start with and you are willing to chance the "Death Kit", go ahead... but at least vacuum it down, and consider a new receiver-dryer! If you're going to spend significant money on a new compressor, then do it right. I've been working on my own a/c for 20+ years and have never been tempted to take the shortcut even when parts and refrigerant were cheaper.
Just my 2 cents worth