Hello @Ajay Chauhan ,
Welcome to Microsoft Q&A Platform. Thank you for reaching out & hope you are doing well.
I understand that you would like to know if the cross region load balancer is a part of data path from end user to regional resources.
Yes, cross region load balancer is a part of data path from end user to regional resources. But when I say that cross region load balancer is part of the data path/flow, I mean the Global Public IP of the CRLB which is any-casted/advertised to the participating regions and not some instance of the load balancer (Azure load balancer doesn't have a concept of instance at all, it works on Software Defined Networking (SDN) architecture).
More information on how Azure software load balancer works, and its architecture can be found in the below blog:
https://techcommunity.microsoft.com/t5/networking-blog/notes-from-the-field-microsoft-sdn-software-load-balancers/ba-p/339800
Global Public IP for Standard Load Balancer will provide an anycast IP that will be announced from all Azure regions (as well as edge locations).
Anycast is a network addressing and routing method in which incoming requests can be routed to a variety of different locations or nodes. Anycast typically routes incoming traffic to the nearest data center with the capacity to process the request efficiently. An Anycast Global IP address is replicated between all the participant regions.
Traffic started from a client hits the closest participating region and travel through the Microsoft global network backbone to arrive at the closest regional deployment.
The traffic for cross-region load balancer goes as below:
End user --> Global VIP in the closest participating region (this is where the global public IP of CRLB is being advertised to) --> backend regional load balancer.
So, the flow would look like below:

Home region is where the cross-region load balancer or Public IP Address of Global tier is deployed. This region doesn't affect how the traffic is routed. If a home region goes down, traffic flow is unaffected.
The home region is not part of the data path. The home region is only used to store the configuration data or metadata of the cross-region load balancer. And hence when the home region is down, you cannot make any configuration changes to the cross-region load balancer, but the traffic remains uninterrupted.
Kindly let us know if the above helps or you need further assistance on this issue.
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