Sorry for delay reply, I mean living checking username for registeration. if there is a live checking username in login, that would be much like potential attack:) what you think? anyway, here is what I am thinking.

1# For any other system, both live checking for signup and signup, and redirect users to previous page on login are very very basic function, we experience a lot every day, but I do not understand why there are millions of plugins for wordpress, but there are selldom plugin for live checking plugin, maybe because of heartbeat? I am not technical background, and I think there is way to solve, for example, limit heartbeat frequecy built in live checking plugin.

2# Secondly(I think this is the most important point), Mulisite are become more and more popluar, it is much convinient for deloy different application in different subsite, and the advantages is reduce potential confict among plugins and plugins with theme, and reduce the time we work. For example, any comercial and business company need a site including brand main site, shop site, community site, login site(with CRM), so I am using Multisite-auth (https://wordpress.org/plugins/multisite-auth/) which assign a single sub site to login, it is much helpful to manage sites. so it would be great if live checking plugin support Mulitiste-auth, since multiste-auth use wordpress standard. please let me know whether it is possible, thanks.

3#Strictly, login and regisration is quite indepedent to user manangment. There are lots of uer manange plugins which provide login and profile, and build-in login, like Ultimate member, buddyrpess, Wp user manager, but no one built in live checking username for registration, and there are indenpent login and registration forms in all of those plugins. the plugin of LWA( Login with Ajax) is great, but no live Checking, since the registertion and login of LWA is jut comply with wordpress default or standerd. so I think the live Checking should comply with wordpress default mechanism, that would be also helpful to spread usage.

So, three points:

1.live checking username in registration.
2.Mulisites-auth support
3.login with Ajax support.

Thanks, and have a nice day.
Alex