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
When you pick an option in the blockchain wallet, when you're making a transaction, to lower the fee you'd see a notification that your transaction may be confirmed far slower or not at all if your fee is too low. So yeah, transaction speed is dependable on the amount of fee you're paying.
Lower the fee = faster confirmation, and vice versa.