I have the same problem. I have a clean install of Windows 10 (final release) and everything looked fine, but when I try to write to an external USB hard drive, I get write errors and then the hard drive freezes up. I first thought it was a bad hard drive, but I tried the drive out on several other machines and it works just fine. It even works fine on other Windows 10 machines that don't use the Asmedia USB eXtensible Host Controller. Its funny though, because the controller seems to work fine on the externally powered 3.5' external drives that I have, but it seems to fail on all of my 2.5' bus powered portable drives.
All of these same external disks worked just fine on this exact machine when it was running windows 7 Pro last week. I thought that maybe the bus powered drives weren't getting enough power, so I used a dual USB cable to get the power from 2 USB ports. That didn't seem to help. The externally powered disks were the Seagate Expansion 5TB Desktop class disks (and these seem to work fine).
** Update: it seems Windows 10 'Build 10586' November Refesh fixes buggy drivers, so its recommended to update Asmedia Windows 10 Drivers for x64/x86 Asmedia ASM1040. Samsung Kies Drivers Windows 10.
The bus powered disks were 2 WD passport disks, and 2 Seagate Backup Plus 4TB portable disks (and all of these failed). I don't get any indication of driver problems in Device Manager. Anybody have any suggestions? Dell Xps 8500 Drivers Windows 10 on this page.
I currently use: ASMedia ASM1042 USB 3.0 xHCI Controller  Vista-8.1 x86/x64 [Asmedia ASM-104x107x Drivers Version 220.127.116.11 WHQL] DriverVer=, 18.104.22.168 on my Win10 x64 Pro build 10240. Have you tried to update controller firmware?
At least two are available: 1) Asus_USB3_FW_Update_Tool_v30  (file with-auto firmware update v120816_02_02_6D for ASM104x [asmedia xhci firmware upgrade tool ver 22.214.171.124]. This one can be flashed on Windows. 2) Additional MS-DOS ASMedia_USB3_ASM104x_FW_130201 . This one can be flashed via DOS.