Actual “labor” has a strong social conservative streak, especially in regards to migration.
Notice the protests by farmers, truckers, etc.
The progressive left in America is formed in heavily globalized universities. They are 100% in support of globalization, as they are beneficiaries and producers of it.
I don’t know where the “heart and soul” of the European progressive left is. Other than Spanish socialists, I don’t think there are many.