If this is truly going to be in the cloud, i.e. on a vendor's hardware, somewhere, keep in mind that the licensing might get a little tricky..technical-wise. It's not that that is can't be done, but how you have users access the licenses for Desktop might matter. I remember talking to esri staff at the UC 2016 about this.
Do you manager the Desktop licenses now? If so, you probably know a bit about the single-user licenses (being authorized to a machine) vs. concurrent-use license manager where the users have to have access to the licenses. Both work well, unless something happens ....ie a "machine" has an issue and gets rebuilt before deauthorizing....or a slew of other things. Sometimes the only way to recover is to get tech support and your customer rep involved. They can help restore things, but not if it a regular issue. With the cloud, where I think things like IP can change, could this be an issue? I have not done any cloud testing, so really don't know.
So, this is just a heads-up to talk to esri or someone in the know about how to set up the licenses if you go this route. Might not be a big thing, but better to know.
Of course, you could always go the ArcGIS Pro | ArcGIS for Desktop route......licensing can be handle thru ArcGIS Online....i.e., already in the cloud. But keep in mind, not all functionality is there yet, and it is a completely different look/feel, etc.
