This is "probably" because Windows will cache the file you're copying into RAM and so initially, the transfer rates are quite fast. But this speed degrades over time.
1 MB/s does sound real slow, and to improve it you could use a faster port eg.USB3 or a newer and better USB drive.(I'm making this assumption that you're copying to a pen drive or something, as an HDD generally provides really high speeds). I got a new pen drive and it gives me 6 MB/s with USB2. So you can see the difference it makes.
Edit:
If that's the case, don't copy to the same drive. Such a problem can only be solved if you copy to a different drive.
Here's an explanation of why:
See, if you're copying to the same drive, the read/write head needs to first copy from one part of the drive, then reposition, then write. And it takes time to reposition, and that's the reason it is slow.