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Yes, your passport should be stamped when entering or leaving the Schengen Area.
The Schengen Border Code - Article 11 states that all 3rd country nationals should be systematically stamped.
Article 11(3) gives a list of exceptions:
Other exceptions for 3rd country nationals, who are residents, are not listed in this article.
British citizens will receive a residence permit based on the Withdrawal Agreement Article 18(4).
Any previous issued residence card must be replaced with this document and became invalid after the 31st of December 2020 (and with it the Freedom of Movement).
Do I continue to have the possibility to move freely within the European Union?
No, this possibility ceased to exist when the United Kingdom withdrew from the European Union β unless you fulfil the conditions under the Withdrawal Agreement in a number of countries, in which case you can also assert your right in these other countries. Please note that the procedures and deadlines may be different in other Member States. In addition, of course, if you are also a citizen of another EU or EEA state, you can continue to assert your mobility rights.A further, limited possibility for mobility within the EU exists if you fulfil the conditions for a permanent EU residence permit or for the EU Blue Card. You can apply for these rights of residence for third-country nationals even if you have an entitlement under the Withdrawal Agreement.
In combination with your passport, your new residence document enables you to travel within the Schengen countries for up to 90 days within a 180-day period. However, it does not allow you to work in or move to other countries. To do that, you require permission from the other country concerned.
Your residence document does not constitute any form of entitlement outside the Schengen countries. That also applies to the EU member states that are not Schengen states, i.e. Bulgaria, Croatia, Romania and Cyprus. Ireland has special rules for UK nationals.
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