Routing all external email via your on-premise environment from Exchange Online

In order for you to route all external email from Exchange Online to your on-premise Exchange server(s) you must firstly ensure 'RouteAllMessagesViaOnPremises' is set to true.

This has to be performed via the shell - so we should firstly connect to our Exchange Online tenant:

$session = New-PSSession -ConfigurationName:Microsoft.Exchange -Authentication:Basic -ConnectionUri:https://ps.outlook.com/powershell -AllowRedirection:$true -Credential:(Get-Credential)

Import-PSSession $session

and set it with:

Get-OutboundConnector | Set-OutboundConnector -RouteAllMessagesViaOnPremises $True

and re-configure the send connector on the Exchange Online tenant so that it includes '*'.