Timeframe for Emigration

Dear list members,

I have located information that states that my ancestors received permission
to emigrate from Westphalen in 1851. I am trying to narrow my search for
their arrival in North America and find a date of death for my
ggg-grandmother who died a short distance from landing.

After people received permission to emigrate, did they leave shortly after
receiving this permission, or did this consent expire after a certain amount
of time?

Any advice or direction will be greatly appreciated!

Carolyn Reif