I believe there are two different kinds of oil base "stains". The ones which are an oil with pigment in it and the ones which are closer to a varnish with the stain pigments added to it.
The varnish type will not penetrate the wood much, if at all. Additional coats will darken it slightly but this is because the total coating is getting thicker.
Many of these varnish stains also become hazy and tend to dull the grain and curl of the wood.
They are usually glossy finishes. Whether you like glossy finishes is entirely a matter of personal taste however some people do not like shiny gunstocks.
The oil/pigment type will penetrate the wood with the first few coats however the wood will become saturated and refuse to absorb more after a very few coats. This results in a condition where more coats will not change the color appreciably.
Another thing about these oil based "stains". Because they penetrate the wood they are almost impossible to sand off and because they are oils, they will prevent the (IMO) better alcohol or water base stains from working if you decide to use them later to darken or redden the color.
In other words, what comes out of the can is what you will end up with without the flexability to modify it later.
The alcohol based stains on the other hand can easily be thinned and multiple coats applied. This not only allows you to darken the color but to use other tints of stains to change the color.
If you already know what stains are needed to get some special color, you can mix the stains and store it for future use.
The water based stains will cause the woods grain to rise so whiskering the stock several times before starting the staining process is necessary.
The alcohol based stains do not raise the grain so usually one good whiskering will suffice.
Many people on this site like the old acid type stains like Aqua Fortis or the chemical stains however IMO, for the most part they are sometimes difficult to control and you never know what you will end up with until you've heated them.
Some of the chemical types will take on a green tone after a few years.
For this reason, I tend to avoid the chemical stains and use the alcohol or water based stains.