I have a partner/family member who is planning to join me in Austria in future, will my residency have a bearing on their right to live and work in Austria?

Yes it can.
In 2021 you must apply either for the Article 50 EUV cardAufenthaltstitel "Artikel 50 EUV" The Aufenthaltstitel "Art 50 EUV" is a residency title to be issued to British citizens from the end of the transition period. Its name is derived from Article 50 of the Treaty on European Union. The "Titel" refers to the document or card itself, and not the right, or permission that it confirms. under the Withdrawal AgreementWithdrawal Agreement The Withdrawal Agreement sets out the terms of the UK’s withdrawal from the EU and provides for a deal on citizens’ rights. It sets out a transition period which lasts until 31 December 2020. During this time you can continue to live, work and study in the EU broadly as you did before 31 January 2020.   If you are resident in Austria at the end of the transition period, you will be covered by the Withdrawal Agreement, and your rights will be protected for as long as you remain resident in Austria.   Any rights that are not covered by the Withdrawal Agreement will be the subject of future negotiations.   https://www.gov.uk/government/publications/new-withdrawal-agreement-and-political-declaration  , or for one of the titles to be able to stay as a 3rd country national3rd country nationals Third country nationals are citizens of non-EU/EEA countries. Residency and naturalisation procedures for third country nationals are more complicated than for citizens of EU Member States or EEA signatory states.. The 27 EU nations and the UK wouldn’t have argued for 2 years to get this deal if it were worse than the alternative. In our eyes it’s a no-brainer. The Article 50 EUVAufenthaltstitel "Artikel 50 EUV" The Aufenthaltstitel "Art 50 EUV" is a residency title to be issued to British citizens from the end of the transition period. Its name is derived from Article 50 of the Treaty on European Union. The "Titel" refers to the document or card itself, and not the right, or permission that it confirms. wins hands down.

Withdrawal AgreementR-W-RRot-Weiss-Rot, Daueraufenthalt EU and Blaue Karte EU Titles The EU Blue Card is issued in the shape of a bank card and is a combination of residence permit and employment permit. If you are the holder of an EU Blue Card, you are entitled to temporary settlement and to employment with a specific employer. It is issued for a validity period of two years. If you held an EU Blue Card for two years, you may apply for a Red-White-Red Card plus, provided you were employed for at least 21 months during the previous 24 months in a job matching your qualification. The Red-White-Red Card  is a combination of residence permit and employment permit. It is valid for two years and entitles you as the holder to settlement and employment with a specific employer. If you change your employer within the first two years, you have to apply for a new Red-White-Red Card. The Red-White-Red Card plus   entitles its holder to settlement and unrestricted employment. If you are the holder of a Red-White-Red Card plus, you may change your employer at any time without having to apply for a new permit. If you held a Red-White-Red Card for two years, you may apply for a Red-White-Red Card plus, provided you were employed in accordance with the eligibility requirements for a minimum of 21 months during the previous 24 months. Family members of Red-White-Red Card holders and of holders of an EU Blue Card may apply for a Red-White-Red Card plus right away.  
Who can come?core familycore family Direct and close family members. Spouse, registered partner, children under 21 (or over 21 if still dependent), dependent direct parents or grandparents; Adopted children are treated as natural children. (spouse/partner, children, grandchildren, parents, grandparents…)spouse/partner and unmarried children
Language testnoneA1 – may not be more than 1 year old.
After 5 years B1 is required for longer residence
Income test(nominally €1,375 for couples but unlikely to be applied), or a job at least 5.5 hours a week€ 1,524.99 for couples
Duration of permit and cost5 years (€50), then permanent residence is awarded (€70)3 years (€120). After 5 years (€120)
Who can get one? Anyone resident in Austria before 01.01.2021Only highly qualified, or people in a required profession.
Comparison of Withdrawal Agreement against Rot-Weiss-RotRot-Weiss-Rot, Daueraufenthalt EU and Blaue Karte EU Titles The EU Blue Card is issued in the shape of a bank card and is a combination of residence permit and employment permit. If you are the holder of an EU Blue Card, you are entitled to temporary settlement and to employment with a specific employer. It is issued for a validity period of two years. If you held an EU Blue Card for two years, you may apply for a Red-White-Red Card plus, provided you were employed for at least 21 months during the previous 24 months in a job matching your qualification. The Red-White-Red Card  is a combination of residence permit and employment permit. It is valid for two years and entitles you as the holder to settlement and employment with a specific employer. If you change your employer within the first two years, you have to apply for a new Red-White-Red Card. The Red-White-Red Card plus   entitles its holder to settlement and unrestricted employment. If you are the holder of a Red-White-Red Card plus, you may change your employer at any time without having to apply for a new permit. If you held a Red-White-Red Card for two years, you may apply for a Red-White-Red Card plus, provided you were employed in accordance with the eligibility requirements for a minimum of 21 months during the previous 24 months. Family members of Red-White-Red Card holders and of holders of an EU Blue Card may apply for a Red-White-Red Card plus right away.   residency for family members

Austrian migration website FAQs

There is a calculator here to see if you qualify for any of the third country national3rd country nationals Third country nationals are citizens of non-EU/EEA countries. Residency and naturalisation procedures for third country nationals are more complicated than for citizens of EU Member States or EEA signatory states. (3CN3rd country nationals Third country nationals are citizens of non-EU/EEA countries. Residency and naturalisation procedures for third country nationals are more complicated than for citizens of EU Member States or EEA signatory states.) cards.

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