Different folks use different products.
Some use Obenauf's which is a beeswax/propolis/plan oil product developed by a wildland firefighter to help protect and restore leather items (boots, gloves, jackets, etc) exposed to ash, oils, water, fire suppresants and retardants, dirt, etc.
Others use Snoseal, another beeswax based product used for "sealing" leather garments and boots against water intrusion, etc.
The alleged downside of those two products is that they can cause leather to dry out over time. Why? I don't know, I just know that it has been repeated on the interweb without any scientific reasoning for a long time. It may even be true.
With mink oils, they come in various flavors. Some are pure mink oil (rendered mink fat). Others have "other stuff" added to the oils, supposedly making them "better" than pure mink oil for waterproofing and application ease. It can also "go bad" or turn rancid over time. Don't expect a large container to stay good for years and years.
Use of any of them WILL darken the leather to which they are applied. Mink oil tends to darken less, if only due to its original source - mink pelts, but darkening will still occur, although more slowly. If the leather is already dark due to use/oils/dirt, no big deal. Brand new NIB, they will do their advertised jobs well, but the darkening WILL occur.
For really dry leather, say like what's on 30 to 80 year old stacked leather handles. The beeswax products don't really penetrate well for "restoring" dry cracking disks, but they do help condition the surfaces, help kinda hold them together like a sticky surface glue. You can achieve some penetration by dint of a LOT of elbow grease slowing rubbing in coats over time.
Mink oil penetrates better, which is great for really dry leather BUT, I have found that it is REALLY REALLY easy to overdo it and end up with perpetual seepage of oil from the leather it is used on. It will make the leather act like an oil sponge and soften it up, not really good for knife handles which take a lot of abuse. A little goes a LOOOONGG way.
Some folks advocate the use of neatsfoot oil. This is made from shinbones of cows (and sometimes horses). I have never tried it for handle restoration/conditioning. It is good for waterproofing, but due to an acidic nature reportedly causes stitching to decay/deteriorate. That may be true for "natural" thread, but highly doubtful for synthetic threads, unless the synthetic material is REALLY susceptible to acid effects. The difficulty with neatsfoot is finding good quality stuff. So much of it is adulterated with other things, including lard. It too can go rancid over time. The best is "pure" neatsfoot oil, but even that tends to have adulterants in it. True pure neatsfoot oil containing nothing but cow shinbone oil is damn near impossible to find.