[SalesForce] Old and New Values of Parent fields with Re-parenting on Child Trigger

First, do you need those fields on the child object in readonly mode ? if so you could use a formula field.

otherwise if you are looking to only auto-populate the child fields with the same values as parent values when you click on the new button, I suggest that you take a look at this blog poste

I am not sure if you could use the field set directly to compare but you can use the sObject's get method to do the job

set<String> fieldSet = new Set<String>();
//dynamically get the fields from the field set and then use the same for comparison in the trigger. 
for(Schema.FieldSetMember fields :Schema.SObjectType.Account.fieldSets.getMap().get('yourFieldSetName').getFields()){
    fieldSet.add(fields.getFieldPath());
}
for(account a: trigger.new){
    set<String> changedFieldSet  = new Set<String>();
    for(string s: fieldSet){
        if(a.get(s) != trigger.oldMap.get(a.Id).get(s)){
            changedFieldSet.add(s);//adding fields whose value changed
        }
    }
    if(changedFieldSet.size()>0){
        //do something
    }
}