It at least used to depend on your luck and how merciful Microsoft happens to be feeling. Back in the day I reactivated OEM Win7 just by clicking the button after moving from AMD to Intel, but reactivating OEM Office 2010 required a phone call. However, it worked. Lots of people have been able to reactivate the OEM version by making the call, even though it technically shouldn't be possible. However, I don't know if those dubious sites that are selling the OEM versions for candy money have changed MS's attitude. It could have happened. I did pay ~100 euros for my OEM Win and Office, after all, buying from regular brick and mortar stores. I do have a gut feeling things might have changed, which is why I got a retail version of Win10 already for my previous PC.
They changed how activation is done with Windows 10. The Product Key and activation is tied to the motherboard now. With Windows7, activation was tied to the product ID. When I switched from Intel to AMD, I ended up getting a new Product Key for free.... I called MS and ripped the support people a new one because I had a Retail Win8 key prior to the forced "upgrade" to Win10 and the digital OEM license wouldn't let me re-activate Windows. I was on the phone with them for almost 2 hours in total literally yelling at MS support staff.....
This last motherboard switch was the easiest. Mine wouldn't activate until I used Caesar's method with I changed hardware recently option then it was fine.