Moving a lead in from another campaign does not count as a new lead on its own. The New lead trigger and the campaign notification New lead only run when all three of these are true.
In the Move leads dialog, turn on New status.
Pick the first status of the target campaign. Only that status counts as a new lead there.
Turn on Trigger automations. The switch is off by default. It only shows when New status is on.
Without New status, the lead keeps its current status, or you map the statuses. The trigger does not fire.
If the lead lands in any other status, New lead does not fire either. Use the Status change trigger instead. It runs for every status you pick, as soon as Trigger automations is on.
This applies to both:
the email automation with the New lead trigger
the campaign notification New lead to client logins