We can estimate the most probable dates for the start of use for each design and most variations, but unfortunately, not all. The following are a best guess based on a number of sources, but are subject to correction.
We have listed the introduction dates, but hesitate to estimate the stop date for any particular variation because, of course, there would be supplies of badges left with stores & groups that would be used until they ran out. So one variation might be around for several years after the design officially changed.
In addition, some badges were produced locally by hand when official supplies were too hard to get, or not available. We have absolutely no way to date these except to say that they fall within a general design type.
Generally for Scouts:
The same basic typology works for other sections with minor variations. For example the last Cub proficiency badges, before the new program was introduced in 1965, had a rolled edge; however, the Scout's before 1968 did not.
There are also some odd ball items, such as the early multi-stage badges and the Scout Experimental Scout Program badges which were different and require separate dating.
In most cases it will be reasonably safe to use the end of the introduction of a new variety as the end date for the previous one but recognizing that this is not completely accurate. The exception would be the metal badges as both metal and cloth were official for a while.