Sub mesa's has probably hit the nail on the head.
If your external hard drive itself can only accept data at 60MByte/sec, then that's going to be the limiting speed no matter how fast USB 3 is. 60MByte/sec is a pretty good speed for an external "portable" unit that uses a 2.5" drive internally. And if it's an older 3.5" drive or if you happen to be transferring files to the last portion of the drive where performance is the poorest then it's probably around the rate you can expect.
I assume you're reading the transfer rate from the progress window that's shown when you drag-and-drop a file? The reason it appears to be faster at first is that Windows caches writes in RAM - so the transfer rate you see initially is the speed to copy from the internal drive to the RAM cache. Of course the RAM cache is ultra fast, so the bottleneck at that point is the internal drive. Once the RAM cache fills up then the bottleneck shifts to the external drive and the transfer rate that's shown will drop to the speed at which it can accept data.