trigger populateChildObject on Lead(after insert, after update){
	// to create new child record whenever a new lead is created
	if(Trigger.isAfter && Trigger.isInsert){
		List<customObject__c> objList = new List<customObject__c>();
		for(Lead l:Trigger.new){
			customObject__c tmpObj = new customObject__c();
			tmpObj.fieldA__c = l.fieldA;
			tmpObj.lookupField__c = l.id;
			// copy other required fields as well;
			objList.add(tmpObj);
		}
		insert objList;
	}
	// to update existing child object record based on new values on lead object record
	else if(Trigger.isAfter && Trigger.isUpdate){
		//get map of lead having list records of their respective childs
		//considering the fact that one lead may have several childs.
		// even in case on single child, the lofic would work fine
		Map<Id,Lead> leadMap = new Map<Id,Lead>([Select id, allOtherLeadFields, (Select id, allOtherCustomObjFields
									From CustomObj__c) From Lead Where id IN:Trigger.New]);
		List<customObject__c> objList = new List<customObject__c>();
		for(Lead l:Trigger.new){
			List<customObject__c> tmpList = new List<customObject__c>();
			// get list of customObject__c records where id of their parent lead matches current lead's id
			tmpList = leadMap.get(l.Id);
			for(customObject__c c:tmpList){
				//update or  populate fields on customObject__c based on your update logic
				// e.g: c.fieldD__c = l.fieldD__c;
			}
			//add all entries of tmpList in objList (objList will be finally updated)
			objList.addAll(tmpList);
		}
		update objList;
	}
}