I think it doesn't depent in the fee but there are no fee transaction that is instant fast in sending bitcoin into others.,,
I think it all depend in the website or wallet you are using it all depend in there system if they have a fast system the transaction will be smooth.,
We Know of many sites that make a transaction faster than the other and also there are many sites that make a transaction take 3-7days
Really, it's the opposite of what you and the OP said. it depends on the transaction fees. The higher the transaction fees, the faster the transaction reaches its destination. Maybe there are many factors (like what AngryDwarf have said above) that determine how fast the transaction arrives, but the most obvious factor that concerns both me and you is the fees.