The Economist is being vague, probably because some argue that freedom of movement has ended (using a future post-transition period), others that it hasn't changed much (using the present context). In fact most things will remain similar to what they are now during the transition period and likely a few years afterwards. There are however new sources of friction for whomever wants to stay long term in the UK.
As for the customs union notice that it has no direct relation with the freedom of movement. Turkey is in the customs union. You probably mean the single market which requires the 4 freedoms to be respected. Not the purpose of your question but you can learn more about this here (in relation to the draft deal).
NOTE: This is a big document and I've only read a few bits and pieces. I've quoted specifically those articles because they seem to have little or no ambiguity. I'll try to update this answer if I learn something.
Indeed the Free Movement will continue
for at least 5 years after the transition period during the transition period (refer to EDIT1 for my source on this).
On the Article 14 of Draft Agreement on the withdrawal of the United Kingdom of Great Britain and Northern Ireland from the European Union rules are set for residents in the host state:
Right of exit and of entry
Union citizens and United Kingdom nationals, their respective family members, and other persons, who reside in the territory of the
host State in accordance with the conditions set out in this Title
shall have the right to leave the host State and the right to enter
it, as set out in Article 4(1) and the first subparagraph of Article
5(1) of Directive 2004/38/EC, with a valid passport or national
identity card in the case of Union citizens and United Kingdom
nationals, and with a valid passport in the case of their respective
family members and other persons who are not Union citizens or United
Five years after the end of the transition period,
the host State may decide no longer to accept national identity cards
for the purposes of entry to or exit from its territory if such cards
do not include a chip that complies with the applicable International
Civil Aviation Organisation standards related to biometric
No exit visa, entry visa or equivalent formality shall be required of holders of a valid document issued in accordance with Article 18 or
Also things would look pretty similar for EU citizens the foreseeable future:
Status and changes
The right of Union citizens and United Kingdom nationals, and their respective family members, to rely directly on this Part shall not be
affected when they change status, for example between student, worker,
self-employed person and economically inactive person. Persons who, at
the end of the transition period, enjoy a right of residence in their
capacity as family members of Union citizens or United Kingdom
nationals, cannot become persons referred to in points (a) to (d) of
The rights provided for in this Title for the family members who are dependants of Union citizens or United Kingdom nationals before
the end of the transition period, shall be maintained even after they
cease to be dependants.
But some of the articles (see 18) also set out the rules and guidelines for permanent residence (after the transition). This increase in bureaucracy will necessary create friction for people intending to stay in the UK in the long term.
EDIT1 (16, November, 2018): As @phoog point out these articles refer to citizens in the host state (UK in this case). I didn't manage to find a clear assertion of freedom of movement rights in the draft (but again I haven't really read the document either). For now I can only quote the EU press release on the subject:
The transition period is set to end on 31 December 2020, taking into
account the initial request from the UK for a transition period of
around two years, and making it coincide with the end of the current
long-term EU budget (the Multiannual Financial Framework 2014-2020).
During this period, the entire Union acquis will continue to apply to
and in the UK as if it were a Member State. This means that the UK
will continue to participate in the EU Customs Union and the Single
Market (with all four freedoms) and all Union policies.