To the OP and others...

Out of curiousity, are you guy flying the segments of your multisegment AONE/DONE e-tickets within a short period of time and on multiple carriers that are different from that of the ticket issuing carrier?

I am suspecting that your 10-20 minute check-in time for each segment may have to do with the carriers and/or ground handling agencies using "working copies" vs. "live copies" of e-ticket records (when the e-ticket is issued by another airline).

As in a paper ticket, the segments on an e-ticket must be flown in the order as shown on the ticket.

Say you have an e-ticket issued by AA with segments on CX and BA in addition to AA. (In this hypothetical situation, this e-ticket will have only 3 segments, one with AA, then the next with BA, and next with CX)

JFK-LHR AA
LHR-HKG BA
HKG-JFK CX

The E-ticket record (referred to as "ETR") when first generated will reside in AA's system, and "working copies" sent to BA and CX. BA and CX will each take "control" of the flight coupons involving their flights a certain number of hours before the flight (usually 72 hours).

Say you have just flown the JFK-LHR AA segment, In AA's system, the "live copy" of the ETR will then be updated to show the AA segment as "FLOWN" (or "CKIN" if you have just checked in but not flown the segment yet).

However, once you get to LHR and decide to fly the BA segment segment to HKG, when checking in, the BA agent pulls up the "working copy" of AA's ETR from their system and sees that the previous JFK-LHR AA segment is marked as "OPEN FOR USE" along with the other segments. On first thought, he/she may have thought that you didn't fly the JFK-LHR segment and tried to board at LHR. But then what they then would do is to "connect" to AA's system and "refresh" the ETR to see what exactly has happened. They see that your JFK-LHR AA segment has indeed been marked as "FLOWN". This gives them the OK to then check you in for the LHR HKG BA segment (which is then updated to "CKIN" after check-in and then "FLOWN" once airborne).

Once you get to HKG, something similar happens again... When the CX agent pull up the record (yes, it's the "working copy" on their system!!) of the AA E-ticket, they will see that your JFK-LHR AA segment as "FLOWN". After all that time, the "working copy" in CX's system would have been synchronized with that of the "live copy" in AA's system, right? Not exactly, your LHR-HKG BA segment is still showing "OPEN FOR USE" along with your HKG-JFK CX segment.

The CX agent would then have to connect to AA's system and pull up the "live copy" of the ETR so that they can refresh their "working copy" and then your LHR-HKG BA segment would show as "FLOWN" before they can check you in for your HKG-JFK CX segment.

Ideally, after a certain period of time, the "working copies" of the ETR on the various carrier's system will be synchronized with that of the "live copy" on the issuing carrier's system.

If you flew via DXB, then this would have futher complicate things as they use DNATA as their ground handling agent. Think of all the fun their agents will have checking in a FTer who's on an aggressive mileage run on multiple carriers