This article explains how Swavalamban Yojana subscribers were moved to Atal Pension Yojana when APY started in 2015.
Swavalamban subscribers aged 18 to 40 were moved into APY automatically, with the option to opt out. Their aggregator helped them complete the migration. The government co-contribution was capped at 5 years across both schemes. So someone who had got 1 year of co-contribution as a Swavalamban subscriber got it for only 4 more years under APY, and so on. The co-contribution ended after 2019-20, so it no longer affects anyone joining today.
Swavalamban subscribers over 40 who didn't want to continue could opt out and withdraw the whole amount as a lump sum. Or they could stay in until 60 and receive an annuity under that scheme.
Banks collect contributions by standing instruction and must send them to the Trustee Bank account the next day. The Trustee Bank then passes the money to the pension funds as per its regulations.