Hello,
1) 'Host A will receive the local certificate from Host B and will use the CA-Certificate to validate it' --- In simple terms you are right.
2) At no point of time Host sends the CA certificate to other host generally. It only sends the local certificate. The receiver checks whether he has a trust chain built.
e.g. Only if ''CA-Sales - CA-Root'' trust chain is built on Host A and ''CA-Marketing - CA Root'' chain is built on Host B (This generally happens when you load host and CA certificates on device) they will be able to authenticate each other based on their local certificates.
Regards,
Rushi