This function is missing with Asterisk.
Phones do support this function. Aastra for example clearly state that they support the caller ID change during a call. (this parameter need to be loaded through the aastra.cfg file : sip update callerid: 1).
Most numeric telephone systems do have this functionnality.
After patching Asterisk 1.2 with metermaid (directed call pickup) and parking BLF, i think it is one of the last big hole we have inside an Asterisk system compared to traditionnal telephony.
This needs to be adressed fastly. I hope this will be backported to asterisk 1.2, because the testing i’ve done with 1.4 show disfunctionnement with advanced functions (directed call pickup and intercom do not work at all for example).
Last, there seems to be some work to get this working with asterisk 1.4 :