This time of year, mid-October, is a good time to look for caterpillars and chrysalises and bring them into a protected area.
The problem with just letting them stay on plants is that they may not develop properly with the colder temperatures. Also, if they do make it to the chrysalis stage, they are so well camouflaged, that they are almost impossible to find in the spring when you are cleaning up the garden waste.
Monarchs need to fly to Mexico, so as soon as they form the chrysalis, try and keep them in a warmer area so they can develop into the butterfly stage. When it gets over 50 degrees, you can release them outside.
Black Swallowtails over-winter in the chrysalis stage. I like to keep the chrysalises in the garage which is not heated. They will usually emerge in April. If you do find a chrysalis in a outside area that you won’t be messing with, leave it alone.
Giant Swallowtails also over-winter in the chrysalis stage, so if you have them keep them in a cold area over the winter.
I like to raise caterpillars in a fish tank covered with a metal mesh screen. I put paper towels on the bottom. The caterpillars climb to the top and hang from the mesh. I prefer this method since it is easy to sanitize the fish tank with a bleach solution.
Another method is to use a laundry hamper with paper towels on the bottom.