A slow migration would be less risky. Whenever budgeting issues come up along with the cost of software, then offer solutions that save money, one at a time as they arise. Whenever a project comes up with an Open Source solution that can handle the task at hand, mention the savings and ask, "Why spend that on software when we can spend it on something else more useful". There might be new projects that come on board that would call for the purchase of new software and staff. Whenever discussions of trimming back costs come up, ask why the organization isn't trimming the cost of software down so that less trimming has to occur elsewhere, especially when viable solutions work that are completely free. At those times, offer to use the Open Source solution and save the organization money.
When new employees come in, give them immediate access to open source applications, with proprietary software to arrive "as needed" When they need something expensive have them fill out a request for software with the reason they need it, then reply back with a question "Doesn't this software that's installed already do that function?" Eventually, turn these new employees over to help established employees learn the shortcuts of the Open Source way. Give the new employees a badge of honor or employee of the week for doing so. Make them the organizations new front line for innovation.
Lastly, when management asks you, "But why haven't I heard of this software before?" Simply retort back the truth, "Did a salesman try to sale it to you, or did you see it at a Vendors Booth?" ...and "That's because no one ultimately needs us to use this software, besides ourselves, because we are the only ones profiting from it"
Over time, some of the computers in a large organization can be replaced with complete open stack systems, like Ubuntu, to handle nearly everything, while a few others may have mixed applications that are in some state of transition to open source alternatives, as a goal. But, there may be others in your organization that do need to stick completely with proprietary systems. There isn't an open source alternative for everything; pushing too far, too fast back fires.
The amount of time spent learning applications and operating systems is MOSTLY UNNOTICED. This is a huge amount of time we spend, in our free time, getting familiar with new software--it's mostly an unconscious process of learning. The slow migration approach (with occasional leaps) is best, else disappoints will arise later, and with that comes the inevitable recommendations against the products that would otherwise be beneficial to any organization.