How to copy dynamic items and all the translations from one site to another in a multisite and multilingual project ?

Use the Code Reference for the dynamic module provided by Sitefinity for every dynamic module to populate the different fields and know the full data type. 

Sitefinity documentation, Code reference for Dynamic Modules: https://www.progress.com/documentation/sitefinity-cms/for-developers-dynamic-modules#code-reference-for-dynamic-modules

E.g: 

...
...
string carTypeStr = "Telerik.Sitefinity.DynamicTypes.Model.Cars.Car";
  
public void RetrieveCollectionOfCarsAndCopyToTarget()
        {
 
            var providerName = ddlSourceSite.SelectedValue;

            // Set a transaction name
            var transactionName = "someTransactionName";

            dynamicModuleManagerSource = DynamicModuleManager.GetManager(providerName, transactionName);
            Type carType = TypeResolutionService.ResolveType(carTypeStr);  
            
            // This is how we get the collection of Car items in Master version 
            var myCollection = dynamicModuleManagerSource.GetDataItems(carType).Where(d=>d.Status== Telerik.Sitefinity.GenericContent.Model.ContentLifecycleStatus.Master);  // IMPORTANT! Get the Master version only
            

            foreach (var item in myCollection)
            {
                CreateCar(item);  // Creates the Car item in the other site's provider. 
            }
            
        }


  public void CreateCar(DynamicContent item)
        {
            var providerName = Regex.Replace(ddlTargetSite.SelectedItem.Text.ToLower(), @"[^\w\-\!\$\'\(\)\=\@\d_]+", "");  // gets the provider name from the UI element (see code attached) to this article

            var transactionName = "someTransactionName";
            var versionManager = VersionManager.GetManager(null, transactionName);

            dynamicModuleManagerTarget = DynamicModuleManager.GetManager(providerName, transactionName);
            Type carType = TypeResolutionService.ResolveType(carTypeStr);
            DynamicContent carItem = dynamicModuleManagerTarget.CreateDataItem(carType);

            foreach (var cultureName in item.AvailableCultures)  // creates all the cultures that are available for the item
            {


                using (new CultureRegion(cultureName))
                {
                    // This is how values for the properties are set
                    carItem.SetString("Title", item.GetValue<Lstring>("Title") ); 
                    carItem.SetString("Modelo", item.GetValue<Lstring>("Modelo")); 
                    carItem.SetString("UrlName", Regex.Replace(item.GetValue<Lstring>("Title").ToLower(), @"[^\w\-\!\$\'\(\)\=\@\d_]+", "-"));   // generates the UrlName based on the Title
                    carItem.SetValue("Owner", SecurityManager.GetCurrentUserId());
                    carItem.SetValue("PublicationDate", DateTime.UtcNow);
                    carItem.SetWorkflowStatus(dynamicModuleManagerTarget.Provider.ApplicationName, "Draft");

                    // Create a version and commit the transaction in order changes to be persisted to data store
                    versionManager.CreateVersion(carItem, false);
                    TransactionManager.CommitTransaction(transactionName);

                    // Use lifecycle so that LanguageData and other Multilingual related values are correctly created
                    DynamicContent checkOutCarItem = dynamicModuleManagerTarget.Lifecycle.CheckOut(carItem) as DynamicContent;
                    DynamicContent checkInCarItem = dynamicModuleManagerTarget.Lifecycle.CheckIn(checkOutCarItem) as DynamicContent;
                    versionManager.CreateVersion(checkInCarItem, false);
                    ILifecycleDataItem publishedCarItem = dynamicModuleManagerTarget.Lifecycle.Publish(carItem);
                    carItem.SetWorkflowStatus(dynamicModuleManagerTarget.Provider.ApplicationName, "Published");

                    TransactionManager.CommitTransaction(transactionName);
                } 

            }

        }