First of all, unless your new community experience uses Facebook logins, you have the issue of just getting prior community members to log into the new community. Consider that some percentage of your old membership is basically gone. Some are rarely engaged, and some are very engaged. You are only going to be able to move the very engaged in short order.
Second, what are the new features of the new platform that don't exist in your Facebook community? Is it possible to bring these online sooner and frequently cross post to your Facebook community so your members become used to the new experience.
Third, have you engaged your elite members in making this transition? Ideally you are having them help you design the new experience, provide feedback, and maybe start with cross posting content in both sites. This is the most crucial part of the transition. If your best participants follow you to the new community that gives you the pull for everyone else who is active, and the ability to draw new members into the new site.
Once you have successfully moved your elites over, try to move your most popular content to the new experience also.
Then you put the old site into read only if that's possible with a note where to find the new community. Eventually you might choose to delete the group.