It should take longer than "minutes". You really should flush each filter first before finally hooking up to the DI to make sure any initial impurites/debris is washed away.
Yes, Opus. In a perfect world, the block carbon can be rinsed before you insert the membrane. The NHCl2 blasters (or any lose or hand packed carbon) you definitely want to rinse all carbon fines out before inserting the membrane (if you buy a Chloramine Blaster unit from us, we install a rinse valve) . I suppose rinsing the sediment would not hurt, though I am not sure if the extra effort would be beneficial here.