BGP: Advertising same IP Prefix at 2 Locations (Active/Inactive)

Hi,

Assuming we are migrating servers from Site A to Site B and we will have 2 ISPs at 2 different locations advertising the same IP Prefix, with an EoIP tunnel for purpose of migration.

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Site A Router (IP Prefix configured in BGP) - - 1.2.3.0/24 (ISP A)

|| EoIP Tunnel ||

Site B Router (IP Prefix not configured in BGP) - - 1.2.3.0/24 (ISP B)

++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Question:

If Site B Router is not configured to advertise the IP Prefix in BGP setup, will internet traffic still flow to Site B despite having the IP prefix being advertised by both ISPs?

Ideally, Site B should only go 'live' after migration is fully completed and Site A Router will be shutdown following that. I am trying to simplify the process without involving load sharing/redundancy setup.

Please advise. Thank you :)