What happens to my emails when I switch my live.com alias to a Gmail address?

I've been experiencing constant hacking attempts on my Microsoft account. To counter this, I've added a Gmail alias, set it as primary, and removed the original Microsoft alias. Now, I'm wondering if emails will be sent to both my Microsoft account (listed with a Gmail address) and my actual Gmail account. If so, how can I automatically delete the duplicate emails on my Microsoft account?