The GUTI takes the place of the P-TMSI, which is the temporary id used in 2G and 3G communications. The P-TMSI is still needed since LTE networks must “hand off” users to 3G or 2G networks for voice communication. When a mobile device operating on a 3G network tries to contact an LTE cell, the network will send the P-TMSI nunber. The MME uses this number to get the subscriber’s IP address and other information from the serving GPRS Support Node (SGSN). The SGSN supports both radio access and IP access.
Once the user is on the LTE network, his/her device will be allocated a GUTI. Then when the user wants to use voice or has another reason to move back to the 3G network, the device sends a GUTI instead of a P-TMSI.This procedure is called a Routeing Area Update (RAU). The MME is responsible for mapping the GUTI to P-TMSI, and vice versa.
If a user moves from one MME to another, the second MME will contact the first one for user information. Both MMEs will use the GUTI to identify the user.
I hope this should clarify the GUTU in 3G vs LTE.