If you already have a connector's LinkedIn connections, you can add them yourself. The connector doesn't need to do anything. Only Owners and Admins can do this.
Import their connections
Go to Connectors.
Find the connector and select + next to their LinkedIn connections count.
Under Already have their connections?, choose:
Import CSV: upload their LinkedIn connections export. If they haven't exported it yet, send them How to download your LinkedIn connections as a CSV.
Add Manually: paste the LinkedIn profile URLs of the people you want to add.
The Swarm maps and enriches each connection. They appear under the connector and in People once processing is done. There's no limit on how many connections you can import for one connector.
Don't have their connections yet?
In the same window, enter the connector's email under Request via Connector Portal and select Send Invite. They get an email asking them to share their LinkedIn connections and Google Contacts. They don't need a password or a Swarm account.
Importing for teammates who are users
This is only for connectors. If you upload a CSV from your own account, every connection is added to you. Users who can sign in import their own connections from their account.