10k!? Holy cow! 
If you're moving your site, before you actually move anything, you need to do a few things (if you're also moving your domain to point to a new server) 
- Set your domain's DNS TTL down to 5 minutes. Wait 24 hours for that change to propagate. 
- During that 24 hours you can move your files over. 
- Backup your site. Shut down the old site with a "moving" message. You might want to include "if you see this page, update your DNS since you're pointing to the old server" 
- Change DNS/nameservers to the new host 
- Wait 5 minutes, move your database over, as well as any files that may have changed 
- Open your new site, 
- Change TTL values back to the defaults/recommended 
That should hopefully minimize your downtime. If your usage is high, I'd highly recommend going with a managed VPS provider; you'll like have the same issues on another shared host