trigger ContactTrigger on Contact (after update) {
	
    if(Trigger.isAfter && Trigger.isUpdate){
        if(ContactTriggerHandler.executeHandleAfterUpdate){
            ContactTriggerHandler.handleAfterUpdate(Trigger.new, Trigger.oldMap);
        }
    }
}

public class ContactTriggerHandler {
    
    public static void handleAfterUpdate(List<Contact> newList, Map<Id,Contact> oldMap){
        // check if trigger is running in bulk mode because for a single record updated through UI,
        // the owner of task is updated automatically
        if(newList.size() > 1){
            Map<Id,Id> contactIdToOwnerIdMap = new Map<Id,Id>();        
            // check if owner has been changed
            for(Contact con: newList){
                if(con.OwnerId != oldMap.get(con.Id).OwnerId){
                    contactIdToOwnerIdMap.put(con.Id, con.OwnerId);
                }
            }
            // There can be large number of tasks under these contacts, so its always a good idea to 
            // process them in a batch
            Database.executeBatch(new TaskOwnerUpdateBatch(contactIdToOwnerIdMap), 200);
        }
    }
}

public class TaskOwnerUpdateBatch implements Database.Batchable<sObject>{

    public Map<Id,Id> contactIdToOwnerIdMap;
    
    public TaskOwnerUpdateBatch(Map<Id,Id> contactIdToOwnerIdMap){
        this.contactIdToOwnerIdMap = contactIdToOwnerIdMap;
    }
    
    public Database.QueryLocator start(Database.BatchableContext bc){
        Set<Id> contactIds = contactIdToOwnerIdMap.keySet();
        return Database.getQueryLocator('SELECT OwnerId,WhoId FROM Task WHERE WhoId IN:contactIds');
    }
    
    public void execute(Database.BatchableContext bc, List<Task> taskList){
        for(Task tsk: taskList){
            tsk.OwnerId = contactIdToOwnerIdMap.get(tsk.WhoId);
        }
        update taskList;
    }
    
    public void finish(Database.BatchableContext bc){
        
    }
}