I have established the connection using Service Bridge Between two ServiceNow instances and did the incident integration with service bridge remote task definition. For the incident all the fields are mapping except reference fields(for example Assignment group field).
In the provider instance i have group - Demo Group, The same group i have created in the Consumer instance as well, When i created incident from provider instance, the incident got created in the consumer instance, but the assignment group is not mapping.
But we can seen the assignment group has been sent to consumer instance(in remote task of the incident the assignment group is populated but not on the incident form).
For this i have written an advanced transform to map the display value of the assignment group, but still the assignment group is not mapping. below is the script I'm using.
On the remote task in the consumer instance, the Inbound field Assignment Group shows the display value of the group. These Inbound fields are string fields so they can only show one value which is the display value not the value. There is a field which is not on the form called Inbound vars json. This field contains the values and display values of the Inbound fields. You don't need to interact with it directly. The field mappings and transforms will handle this interaction for you.
Your group name is in the Inbound field - Assignment Group. The sys_id of the group is in the Inbound var json. When Service Bridge is writing data to the Incident, it is looking at the value of Assignment Group in the Inbound vars JSON field. This value is the sys_id of the assignment group in the provider instance.
Looking at the screenshot you provided you can see the icon with circle around the i but no value in the reference field. This is because the reference field is holding a sys_id it does not recognize so it displays blank. The value of the assignment group did make it to the Incident but that assignment group sys_id does not exist.
So you need to create an Inbound Transform to get the correct value before the incident gets updated. You started to create an Inbound Transform but the Information message got you. The message does not pertain to your situation. It has to do with tables and fields in the provider which are not present in the consumer. In this case the Incident table and assignment group field are in both the provider and consumer instances.
Here is an example Inbound transform. I added an else for a default value but not required. output.value and output.label should have a value.
 
