Hello Zubair.
So if I understand correctly your topology is something like this:
rootsw secrootsw
Core1 ==== Core2
\ /blocking port
DISTRIBUTION SW
| | |
site1 site2 site3
DHCP users Discover, Offer, Request and Acknowledge. Remember that when a host in site1 sends a Discover packet this packet has as a destination the broadcast address of the network where it belongs to, because this is a broadcast it will go all the way up to Core1 and Core1 will forward the broadcast across all interfaces in the same broadcast domain reaching finally Core2. Core2 replies back with a Offer and the negotiation goes on...
Should be handing out DHCP addresses because the DHCP request was first received on Core 1?
Not necessarily, what if the the link between distribution and core1 goes down, will you move your dhcp pool back to core2 just because now it hits core2 first.... ? ... anyways due to this is a broadcast traffic it will still hit core1 and core2 everytime there's a request.
Regards.
Wilson B.