How timely should data be integrated between EMR and EHR

How timely should the information from EMR integrated with centralized EHR?

There are different types of data that can be integrated between EMR and EHR, such as

1) Patient admission
2) Lab Order/Result
3) Medication prescription/dispense
4) Discharge summary, etc

Should the above information instantaneously integrated with EHR whenever the it is available in EMR? or should it be integrated with EHR only when the patient is discharged  or at the end of patient visit with exchanging of complete summarized information such as CCD?

If the patient is still in one healthcare institution, why is the need to push the data from EMR to EHR? More timely the data is pushed to EHR, there will be possibly more update and correction such as lab result correction. Sometimes due to manual process involved, some of the local update at EMR  may not be propagated all the way to EHR.

If cross-institution diagnosis is required at this stage during in-patient stay, then wouldn't it be more accurate and complete to access the EMR data instead of the subset data held within EHR.

What's your views?