Usually no downtime or glitches. If you upgrade your VPS, services will remain running on your old IP address while DNS propagates to the new IP address.
However, in some rare cases, some complications may lead to downtime. One of the most prolific cause for downtime during the transmission can be sites that have unique IP addresses. Just because your site has unique IP address, you'll face downtime. It's just a small reason of downtime during up-gradation.
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