If the foster mother is white or is herself from an ethnic minority group, the following phrase, ethnic minority group, is probably the best solution, and one least likely to cause offence or misunderstanding.
- She was deeply protective towards her foster children from ethnic minority groups.
a group of people of a particular race or nationality living in a country or area where most people are from a different race or nationality
Google book results for "children from" ethnic minorities e.g. “Bilingual children from ethnic minority groups in the Netherlands are less successful in secondary schools than are Dutch children of the same age.”
The expression also fits in nicely with the OP's second example
- They shunned the ethnic minorities.
If, however, the children fostered are Caucasian but not their foster mother, and if they are living in a predominantly Caucasian country such as Canada—as the OP's belated edit suggests—than the term ethnic minority would be inappropriate because the foster children ‘belong’ to the largest ethnic group; i.e. White people
The term "white race" or "white people" entered the major European languages in the later 17th century, in the context of racialized slavery and unequal status in European colonies. Description of populations as "white" in reference to their skin color predates this notion and is found in Greco-Roman ethnography and other ancient sources.
In the aforementioned scenario, the OP has several options open before them. Once the cultural identity of the foster mother is established the children can simply be described as being ‘white’
- She was deeply protective of her white foster children.
If the OP wishes to emphasize that the foster mother is of a different race.
- She was deeply protective of her racially different foster children
If the OP wishes to avoid the undeniable negative connotations associated with the word racial, and in its place use a single word, Andrew Leach's suggestion would be confusing
She was deeply protective towards her allogeneous foster children
The noun race has to be added in order to clarify. I looked in Google Books and there is not one instance where the expression ‘allogeneous children’ is used. I searched for ‘allogeneous people’ and found nine examples, four of which are from the same source.
Inorodsty, is a legal term used in the Russian Empire in reference to non-Slavic population of the empire. Literally meaning ‘of different descent/nation’, it is sometimes translated as allogeneous (people) (cf. ‘allogenes’) and sometimes as ‘aliens’.
In 1913 the Empire counted 70 million Russians proper, together with 90 million 'allogeneous people'
Communism in Romania in the inter-war and immediate post-war periods was represented by two groups: a) allogeneous people (including Jews) and b) locals (Romanians)
Of course, the idea that the burial was that of a Tei culture member who was buried by the rite of the allogeneous people cannot be excluded.
In mid-13th century it counted some 55,000 inhabitants, of whom 150,000 allogeneous people 30. One century later, it mounted to 900,000, including 65 per cent Romanians.
Before the continuous increase of this utterly allogeneous people in Washington, alured by their inversely disproportionate employment by the District and Central Governments...
The OP's second examples would be
They shunned the allogeneous people.
A reader who is versed in the meaning of allogeneous might well understand that the people or foster children were non-Slavic; or, from different cultures and not of a different skin colour.
In the third example provided by the OP, the overworked and exploited labourers working for the “rich, white mining owner” are of different nationalities made up of ethnic minority groups.