20 gauge slug is a proven deer killer, pretty much any slug, any brand at any reasonable distance will kill a deer.

410 gets tricky, not impossible but you have to make sure you're using the right slug and that your gun has decent accuracy. The Brenneke slug is by far the best performing and most powerful option for the 410. one of the biggest issues with some 410's is poor accuracy with slugs and I've generally equated this with the fact that most fixed choke 410's are have a full choke, especially the cheaper single shot models. most slugs are designed to give the best performance with cylinder bore or improved cylinder.

even with the right slug and good accuracy, range will need to be limited and shot placement will be a lot more critical, especially on big deer. that brenneke out of a rifle basically has the same ballistics as a light loaded 10mm out of a handgun, so just keep that in mind. good luck!