Unless I'm missing something, @SlackHQ makes it very hard to manage multiple slack channels you may be involved with and invited to based on potentially having multiple email addresses.
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So if I get an invite via an email, how do I go to my desktop app and then get that Slack workspace incorporated into my desktop software?
Yep! You can log into multiple accounts with the desktop/mobile apps. Are you logged in on your web browser after accepting the invite? Click the workspace name in the top-left to open the menu -> "Open the Slack app" to add to your desktop app.