Question about multi asterisk and caller ID

Hi all, I have the situation shown in the pic:

I have a central office, where will be reside a call center and 3 satellite offices, every satellite office(A, B and C) has a ISDN PRI (about 20 channels each) connection, but all the operators will be in the call center.

I was thinking about installing an asterisk box in each office (so 4 boxes).
I was thinking if after the call is redirected from a satellite office to the central office the original caller ID will be shown to the operator of the call center, or if it will be lost in the transmission of the call from satellite office to the central office.

Thank you everybody for your kind attenction

nobody can reply this question?
maybe I should post in developers forum?

With proper configuration the original caller ID will NOT be lost.