The use case here, customer is still in Classic, but working to migrate to Lightning by introducing Lightning for pilot or partial users, so the rest of other users still in Classic.
In Classic, Global Actions will appear in Chatter tab.
The actions available and order is following "Quick Actions in the Salesforce Classic Publisher"
If you notice the lower panel at above screenshot "Salesforce Mobile and Lightning Experience Actions", this mean, you can have different global actions available between Classic with Lightning and Mobile.
First, click tool icon at lower right of "Salesforce Mobile and Lightning Experience Actions" to overwrite actions available for "Salesforce Mobile and Lightning Experience Actions".
Now, you should get list of actions in "Mobile and Lightning Experience Actions" exactly the same with "Quick Actions in the Salesforce Classic". From here, you can modify available actions removing exiting actions by drag the action out of panel, or add new action available from "Mobile & Lightning Actions" panel. Let's say we have following actions for Mobile and Lightning Experience.
For users in Lightning, the below screenshot result from changed done in above Global Publisher layout. All Chatter Standard actions will be not available, and the default New Opportunity is not appeared, because this org. have record type implemented in Opportunity.