Is it permissible to perform Umrah during Ramadan on behalf of relatives, and is there a specified degree of kinship or number of Umrahs and their conditions?

If you perform Umrah for yourself, then it is permissible for you to perform Umrah on behalf of your relatives, starting with the closest, provided they are unable to perform Umrah due to old age or an illness from which recovery is not expected. There is no limit to the degree of kinship or the number of Umrahs you can perform on behalf of others.