So if I'm reading your schema right:

D looks up to A with a Master Detail

C looks up to B with a master detail

B loops up to A with a lookup

So to write a trigger to do this AND make sure it's batchable - because you could easily hit limits doing this I would do the following:

1) Create a formula field on object C that references object A's id (example: ObjectAId__c).

The formula would be something like: "CASESAFEID(ObjectB__r.ObjectA__c)"

This will minimize soql queries and make maintenace easier

2) create a batchable trigger as follows:

trigger ObjectC_aU on ObjectC__c (after update){
/*
Processes:
	Set the dates on cousin D objects based on our date
*/
	map<id,date> objectADates = map<id,date>(); //Create a map of Object A ids to object c dates
	for(objectC__c c : trigger.new) {
		ObjectC__c old = trigger.oldMap.get(c.id); //Reference the previous values from the update
		if(old.DateField__c != c.DateField__c) { //If we are changed - then we have an update to do!
			if(objectADates.get(c.ObjectAId__c) == null) {
				objectADates.put(c.ObjectAId__c,c.DateField__c);
			} else {
				//What happens if there are sibling c records that are updating the object with a different date? This if your own business logc that can be added here
				// It may not be needed - just a thought
			}
		}
	}
	
	//Check if we have work to do before moving on... otherwise don't bother using up a soql query
	if(!objectADates.isEmpty()) {
		list<ObjectD__c> cousins = [SELECT id, ObjectA__c FROM ObjectD__c WHERE ObjectA__c IN :objectADates.keySet()]; //Get all the cousing d objects based on the keySet of the map we created earlier
		if(!cousins.isEmpty()) {
			for(ObjectD__c d : cousins) {
				d.DateField__c = objectADates.get(d.ObjectA__c); //Grab the date based on the map
			}
			update cousins;
		}
	}
}

That should do it

PS: On a side note: when you are using multiple MD relationships and are updating cross relationship like this, you will notice that updates will be called multiple times. This is just how MD relationships work (especially when there are roll up fields involved) so it is EXTREMELY important to be very EXPLICIT about your conditions where the updates on the related object occurr - otherwise you will hit stack depth exceptions