I know that a well known and well regarded orthodox Rabbi has o.k’d the use of Kytniot by-producs to the family where I was intending to be for Pessach (till I heard about this). We are all Ashkenazi. The family has small children, some of which might have a milk allergy. The food and utensils are used for all the family and guests. What is the Hallachic basis for this psak? May I eat there?

It is generally accepted that the minhag to forbid Kitniyot includes kitniyot products. There are however certain kitniyot products that are permitted by many poskim - Rav Kook and others - this, when the product is made in a way that ensures that the kitniyot did not come into contact with water before the kitniyot were baked or cooked without water. Soy oil, for example, is made of soy beans that do not come into contact with water and are cooked in petroleum by-products. There are poskim who permit the use of products that include kitniyot as a minor ingredient that is not distinct or separable. This on the basis of R'ma Orach Chayyim 453 1 who permits eating a dish into which kitniyot has fallen during Pesach. You should observe the minhagim of your own family but unless your Rav has given you a different Psak it is permissible for you to eat with this other family non-kitniyot food that has been cooked in their pots. This because kitniyot is batel b'rov.