When a shipment is sent Port A in one Country to Port B in another country what is the detailed responsibility methodology of goods shipped?

As far as i understand from your question, my answer would be like this, if the term of the shipment is FOB-Shipping point it will be the responsibility of the buyer of the shipment the moment it leaves the Port A, but if the terms is FOB-Destination then, it is the responsibility of the seller to whatever happen to the shipment before it reaches Port B.