Two old methods - smoking and shaking
Here are two methods of introducing queens that were taught to me by old beekeepers when I started beekeeping. They probably didn't know the principles of what they were doing, but I suppose by trial and error they found ways that worked.
I have used both these methods on several occasions, mainly when I have been without a queen cage. They are as successful as any I have used. I have no doubt they will be frowned upon by some - probably because they aren't in the books they have read!
For this method to be successful the colony needs to be queenless for at least 4 days. It will probably help if the queen to be introduced has been without food for half an hour or so.
The colony needs to be queenless. I have done it soon after removing the queen, but I think it better to wait an hour or two, so the bees are frantic. It may also be better if the queen is hungry, by leaving her half an hour or so.
I understand that some beekeepers may not wish to try these methods, but I find they work. If you are a bit doubtful, then try with a queen you wish to cull.