When lockdown starts ... no more event cards are draw.
This is not quite true; the actual rule (p 12) is:
Once lockdown has begun, you do not draw an event card when you move onto a tile with a reactivating event icon on it.
Which means (and this is also explained at the top right of page 11) you still draw an event card if you either:
explore a tile with one of the three event icons on it, or
end your turn without having drawn an event card
Is that what you meant, or did you stop drawing event cards altogether? Doing the latter would make for a pretty easy lockdown phase, because the guards and clones would activate far less often (i.e. only when you fail a challenge on their tile).