To answer this question, we need to know how the flow will be triggered.

If your flow is running on an automated trigger, then it will always run in the context of an owner of flow (Who created the flow). So that is who shows up in Modified By field in your SharePoint list.

However, for the manually triggered flows, such as Flow Button, SharePoint “For a selected item” and “For a selected file”, the flow can be configured to run in the context of the user who starts it.

In order to do that, we have to let the person invoking the flow provide their connection (credentials). This is achieved through the Manager Run-Only Users section of the flow properties. This only appears on manual triggers.

  • Click on Add another user in the Manage Run-Only Users tile
    • Add the users with whom you’d like to share the flow.
    • In the Connections used section, click on the dropdown and configure it to be “Provided by run-only user” for the connections you want.