Do you need to clear immigration/customs in Frankfurt before continuing on Lufthansa to Budapest

In Duesseldorf, our EWR-DEU-LHR Lufthansa transfer involved just crossing the concourse, about 20 meters. Interestingly, although we did not exit the secure area (i.e. Immigration booths), they paged us, in German, at the gate to get an check and initialing of our preprinted (in Newark) boarding passes for the DEU departure.

That didn't happen on our return, FCO-FRA-JFK: In Frankfurt, a massive airport, we had to go to the farthest Lufthansa gate, which was listed as 14 minutes walk on the departure boards. As noted, your luggage will be checked through to the final airport and you won't see it or do customs (... walking under the green Nothing to Declare sign, probably) until the last airport.

(The odd routings were to get a good deal on non-revenue mileage tickets.)

I would guess that the airlines allow shorter connections than 1.5 hours, even in that large airport. If your bags are checked through and your flights are code-shared, the airline should make accommodations in case of flight delays, as mentioned above. Of course, don't put valuable of necessary items (like prescription medicine) in checked luggage, in case luggage is delayed.

Two small items - if you have a short connection time try to reserve seats as forward and on the left hand side of the plane. Makes getting off much quicker. 2. If short connection again and you hit a long line at immigration, do not hesitate to catch the eye of some official looking person and show them your ticket. Often there will be a special line for people like you.