Sync local Database with remote REST API

[Leonard]

Hello everybody, I'm having an hard time to figuring this out.
I'm designing a django project based on data retrieved from a remote public REST API that will:

• Retrieve data from a REST API
  
• Saving the data in my models (my local db)
  
• Analyze and process the local data asynchronously and create statistics and trends
  

When a user request the data:

• Check if the local data (models) are in sync with the remote, otherwise fetch the remote data and save them in the local db
  
• Present the data
  

What is the best approach to do this? 
My questions:

1. Should I design my models replicating all the remote API fields?
   
• Can I use JSonField and store data in json format to speed things up?
  
• In this case would it be complex to calculate statistics on the data?
  In other words I can't use the django ORM and foreign keys if I store data in JSON format, right?
2. Where should I call the remote API?
• Considering that the remote data should "replicate" the database models, first I tough to add model methods that call the REST API inside the models
  
• Or I could put them in the manager, considering that when a user needs the data he/she doesn't need to know if they are local or remote
• How do I tell django to grab the remote data and put them in db before presenting them?
• Maybe this should be an asynchronous task? 

Sorry for the complexity, I hope I was clear!

Thanks in advance for your help

[Eron Lloyd]

Has anyone ever come up with a best practices approach for this? Particularly if the remote service is the primary one.