The welfare of the children is always, and completely rightly, paramount.
The feelings of the foster carers will always come second - that is the whole point of foster caring - to do whatever is necessary to give the best possible care to children who have not had that care in the past, for whatever reason.
If the local authority have any concerns whatsoever that a particular set of carers are not the best possible option for a particular child or children, then they are quite right to move them to somewhere they feel is better suited to their needs. It has been recognised for a long time now that a child's cultural and ethnic identity is important and needs to be recognised and supported. It is part and parcel of caring for that child.
If the local authority had concerns about a foster carer's ability to provide a proper diet for a child with allergies or cultural dietary requirements, they would place them elsewhere. If they had concerns about a carer's ability to support an emotionally disturbed child, they would place them elsewhere. If they had concerns about a carer's ability to physically cope with a child with serious disabilities they would place them elsewhere.
In this case they have concerns about the carers' ability to appropriately nurture these children's cultural identity, so they have placed them elsewhere.
The carers should be able to accept that a decision has been made that these particular children are better served in a different environment. Going public rather smacks of this all being about them, rather than about the children.