On EU-Swiss free movement of people
In the EU free movement is allowed per the Citizen's Rights Directive, 2004/38/EC. Switzerland has a similar bilateral agreement with the EU. Article 10 paragraph 1 of that agreement states:
For five years after the entry into force of the Agreement, Switzerland may maintain quantitative limits in respect of access to an economic activity for the following two categories of residence: residence for a period of more than four months and less than one year and residence for a period equal to, or exceeding, one year. There shall be no restriction on residence for less than four months.
From the beginning of the sixth year, all quantitative limits applicable to nationals of the Member States of the European Community shall be abolished.
Per that last part, there are no quotas on nationals from the EU.
What changes with the Institutional agreement (InstA)
The Swiss government hosts a file entitled Institutional agreement between Switzerland and the EU: key points in brief, it summarizes what's in the draft deal. With respect to this question, section 6 on the free movement of persons is relevant. The section starts with:
The InstA applies to the Agreement on the Free Movement of Persons (AFMP), a market access
agreement that is subject to the principle of the dynamic updating of the bilateral access agreements
(cf. the section on developments in EU law). Nevertheless, the Federal Council sought exemptions
from the adoption of EU law ('red lines') in three areas: the accompanying measures, the Citizens' Rights
Directive (2004/38, CRD) and the amendment to Regulation 883/2004 on the coordination of social security systems. The EU's position, in contrast, is that all operators in the single market must be subject
to the same conditions ('level playing field') and that general exemptions are in principle unacceptable.
It is furthermore the EU's view that the dispute settlement mechanism provided for under the InstA could
be used to address and resolve any disagreements about the adoption of EU law on a case-by-case
Only the second of those three (the Citizen's Rights Directive) seems relevant to your question and on that the section says the following:
Citizens' Rights Directive 2004/38 (CRD): from the Swiss perspective, the CRD does not constitute a
further development of the free movement of persons (in the sense of the free movement of workers
under the Agreement on the Free Movement of Persons). Switzerland is therefore of the opinion that it
does not have to adopt the CRD. Various provisions are especially problematic for Switzerland: in
particular the extension of the entitlement to social assistance, the extension of the protection against
expulsion (public policy exception), and the right of permanent residence for persons who have resided
in the country for five years. The EU, for its part, considers that the CRD does constitute a further
development of the free movement of persons.
The CRD is not mentioned in this draft agreement. The InstA thus grants Switzerland no explicit
exemption in this respect. The EU did not insist that Switzerland explicitly commit under the InstA to
adopt the CRD within a specific period of time, as it is required to do in relation to the rules governing
the posting of workers. The dispute settlement mechanism provided for under the InstA would apply in
the event of a disagreement with the EU as to Switzerland's adoption of the CRD. Should the arbitration
panel rule against Switzerland, the terms of Switzerland's adoption, or partial adoption, of the CRD
would have to be negotiated. If Switzerland were to still refuse to adopt the CRD, the EU could decide
to take compensatory measures, which would have to be proportionate. (See Appendix I)
Considering that this is basically an executive summary of the agreement, it seems safe to say that there are no quotas in the agreement as those would have made it into the executive summary.
Therefore, the answer is: no, the new EU-Swiss draft deal does not contain explicit provisions on immigration quotas.