Presently, AMD only supports the newly launched Ryzen 5000 processors on the 500 series motherboards, with support for the 400 series board slated to arrive in January (2021). The older 300 series motherboards including the B350, X370, and A320 aren’t going to support the Zen 3 chips, at least not officially by any of the OEMs, nor AMD.
However, enthusiasts have already managed to boot the Ryzen 5000 CPUs on an A320 board using a custom tweaked BIOS. However, just because a chip is able to boot on a motherboard doesn’t mean it fully supports it. Some platform features, most notably a proper boosting behavior, power profiles, work distribution, etc won’t be ideal on older boards, most notably an A320.
As for the B450 series motherboards, they are going to get a beta BIOS update which will block flashing to the stock or BIOS versions, as well as require proof that you own a 5000 series CPU. Many OEMs are already testing support for the new parts on these boards, and as such the leaked/modified BIOS patches are no surprise.