This is how the EK rules work (haven't checked the exact times - you may want to cross check with the exact schedule):
1. If you take EK 571, which arrives DXB at 13:10, you must connect to EK 131 which departs at 18:50, a connection time of 5hr40mins, inelgible for Dubai Connect in any class.
2. If you take EK573, which arrives DXB at 00:05, you must connect to EK 133 which departs at 09:35am, a connection time of 9 hrs 30 mins, eligible for Dubai Connect in all three classes (and eligible fare basis).
3. If you take EK573, which arrives DXB at 00:05, and you connect to EK 131 which departs at 18:50, a transit time of 18:45 you will not be eligible for Dubai connect, just by virtue of there being an earlier connection (EK571), and even though that flight may itself have been eligible for Dubai connect.