I see the archive link. There is the expected template:

{{Page de discussion wikitexte convertie en Flow|archive=Discussion utilisateur:ԱշոտՏՆՂ/Archive 1|date=2015-10-27}}

and in the rendered form it includes:

"Cette page utilise Flow, un nouveau système de discussion. Consulter la version archivée de cette page."

with link.

I don't manage anymore to see the diff between the old page and the new one.

The page has all these templates:

{{Palette accueil}}
{{grossir|Bienvenue sur Wikipédia, {{BASEPAGENAME}} !|1.5}}
{{NUMBEROFARTICLES}}
{{Grossir|P.S. Vos nouveaux messages seront affichés en bas de cette page et signés par leur expéditeur. Pour lui répondre, cliquez sur sa signature ([[Aide:Discussion|aide]])|0.85}}

Theoretically, these templates are part of the welcome message on French Wikipedia (all text is missing on the example I've pointed). This message is distributed by a bot, and remains sometimes the only message on user talk page.

Flow conversion should have moved the page content to ../Archive 1 and that is not the case here.

Ok, I've found the archive : https://fr.wikipedia.org/wiki/Discussion_utilisateur:%D4%B1%D5%B7%D5%B8%D5%BF%D5%8F%D5%86%D5%82/Archive_1

The page has been moved as expected, but why templates from the welcome message remain in the board description?

In T116787#1761278, @Trizek-WMF wrote:

The page has been moved as expected, but why templates from the welcome message remain in the board description?

It's by design, under the theory that whatever the template is (an article classification, note about conventions for that talk page, etc.) is still relevant to the Flow board.

You mean all templates will remain, because they are considered as relevant by default? Good to know. I'll document that.

In T116787#1880449, @Trizek-WMF wrote:

You mean all templates will remain, because they are considered as relevant by default? Good to know. I'll document that.

Yes, when using the opt-in feature. Specifically, all templates from the old wikitext page's header (before first section): https://git.wikimedia.org/blob/mediawiki%2Fextensions%2FFlow.git/3e10516edb93e80633e4a6fa66fb16e690e23208/includes%2FImport%2FOptInController.php#L81