Hi Oscar, thanks for your reply.
Here is the full dynamic of what happens on our sites:
-site A with WooCommerce and Price Based on Country;
-site B, a blog with in homepage site A’s products took with REST API from site A;
-we update price from site A with WP interface;
-we decide which product to show on site B with a shortcode containing products IDs.
So, to answer your question, we update the pricing on site A and the REST API should return always updated prices on site B.
Thanks for your help.