If there is no issue, what is the proper way to perform this? How long after staining should I wait?
I’ve only found answers to the reverse problem. And not sure if it would be the same answer.
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I'm surprised there isn't already Q&A for this, but I could not find it. (It also occurs to me that I may be assuming you are talking about a final "top-coat" finish on top of a water-based polyurethane style finish. You might be talking about a water-based penetrating stain that you want to top-coat with an oil-based product. Please clarify in the question which one of these you are actually doing.)
In most cases, when re-coating existing finishes you have to scuff up the previous finish. It is no different in this case.
The take-away here is that, in most regards, once fully cured any "coating" finish (i.e., not a penetrating finish) is pretty much the same. That is, the oil vs. water is more about the application of the product and the clean-up, along with considerations for environmentals like VOCs, protective equipment, etc. While there are differences in the final product between these broad categories of finishes, for most of us those chemical and physical differences aren't as important as the wet application.
Yes this is totally fine, there's no issue using waterbased stain followed by a finish with a different base, and in fact it's actually quite normal to do this1.
Often we ask about specific products when they haven't been named in the Question but it doesn't matter in this case — any oil-based finish, including straight oil, blended finishes like "Danish oil" and "Tung Oil Finish" (note: doesn't contain any tung oil!) and all types of oil-based varnish will work equally well over a waterbased stain.
what is the proper way to perform this?
This is a little too broad a query for here. But basically it goes like this:
Note: be very careful if you intend to sand the oil-based finish during application as again you risk revealing bare wood.
How long after staining should I wait?
You should find guidance for the stain you've bought in the product literature, maybe even on the container itself.
If you're not satisfied that the generic advice provided suits your local conditions by all means err on the side of caution and wait a full day. You could wait two days and it won't hurt.
1 Which may explain some of your difficulty in finding info on it, it's sort of a given that you can do this.
2 Because the water in the stain will otherwise raise the grain and you probably won't want to sand the wood (possibly at all) after the stain goes on because this risks sanding through the stain in spots, revealing bare wood. There is much existing advice here on how to pre-raise the grain, as well as elsewhere online.