The pre-requisite to register in a "new to you" EU country's public health insurance as an incoming former EU resident is that you must have been in your original EU country's public health system for 12 months continuously in the last 2 years, or 24 months not necessarily continuously in the last 5 years. The requirement does not depend on whether you hold any mutuelles, or whether you have to pay for public health cover or not, simply that you were covered under the basic public health insurance for your original country.
You can't transfer anything in this situation, you lose entitlement to anything from France as you are simply moving from one country's public health insurance system to another's as you are becoming a resident in your new country. Once you change countries (after being in the French health system for a provable 12 month minimum) you would leave the French system and start in the German system.
Some EU countries public health systems are more difficult to get into for some people, even with EU partners.
So, the system in the EU country that you're planning to use to get into the EU system must actually let you join that system first, of course. You couldn't join the German system for 12 months to transfer to another EU country's system, because Germany wouldn't let you join to start with, if you see what I mean! What you need is an EU country that permits universal public health fund membership with conditions that you can meet. France seems to be ok, as you should be able to join, the other obvious countries are Ireland and Austria which both make it simple to join their public health systems.
For Rixlari:
Actually, since you have German Citizenship but have never worked/lived/been in their health system, you do have grounds for them to let
you join - but your husband does not. So you could simply roll up in Germany and join. Although oddly enough, you couldn't join the German public system on this basis if you have ever been self-employed or freelance in the country you grew up in.
Your husband however, could not join, as he's not German. He could probably be added onto your insurance as your dependent if his income was less than about 425E a month, otherwise, it would be private insurance, or the Basistarif for him - unless he/both of you have been a member of another EU public health system for 12 months. If you join German private health insurance at all, you forfeit your right to join the German public health system.
The German requirement is actually fairly straightforward:
https://www.sozialgesetzbuch-sgb.de/sgbv/9.html
and
https://www.sozialgesetzbuch-sgb.de/sgbv/5.html (see number 13)
Public health systems do vary significantly from one EU country to another. But the common thread is that someone who has been covered for at least 12 months (in the last 2 years) by one EU country's public health system can (subject to the normal requirements for entry to that new system) move into another EU country's health system.
It is absolutely imperative to apply in Germany within 3 months of registering your residence - after that point, the public health system is not compelled to insure you, even if you have 12 months in another EU country's public health system.
The really important thing to remember is that you can't live in Germany without either public or private health cover.
Rather than rely on information posted by me or others, or from single sources, for your own safety, I'd be inclined to investigate more until you are absolutely sure of your situation.
This may also include getting professional-level advice from a versicherungsberator or two - they may not be willing or able to advise you much as they may mostly deal with private health insurance, but they may help you find out about and even register for public cover for a fee. Or if you both agree that can't be done, they might be able to get you into the Basistarif with a reluctant krankenkasse, or find good private health insurance, for a fee of a few to several hundred euros, or they may able to direct you to someone else.
The link I provided to free public support in my first post may also provide help and confirmatory evidence of what would be acceptable for your situation, but you might have to be there to get that support.
I would caution too that there may be significant and far-reaching changes to the German health insurance system over the next few years - if you are not planning to arrive in the quite near future, requirements may well change. There have been strong political rumblings about this for some time in Germany now - the SPD in particular want the health system changed to become more universal. However, their support nationally is not what it used to be. Then again, maybe a deal will be done with the CDU (who they don't really like too much at the moment) which may include this, so that the CDU can try and retain power. Who knows, especially after Merkel goes and AKK holds the reins?
Given freedom of movement is an essential right for EU Citizens, I don't see that the ability to change countries and transfer to a new public health insurance system will change, but the necessary requirements to join your first EU country's system might!
In most countries, the rules sometimes change (often very significantly) over several years. Things may become harder or easier for your particular circumstances, and indeed your circumstances can change over time too! So if you're planning to move say in a few years, sure, get a feel for the requirements now, but check again closer to the time of your proposed move too.