They are still around. Some Puritan congregations sort of 'morphed' into slightly different variations, but many didn't really (though their doctrine and beliefs continued to evolve).

The Congregational Churches are the direct descendants of the Puritan churches.

Because congregational churches have maintained that there is no higher authority that each church reports to than its own congregation, there has been more diversity of doctrine and belief among Congregational churches than among some other more hierarchical churches.

Many Congregational churches adopted a "Unitarian" theology around 1800, and in 1825 the Unitarian Congregationalists separated from the rest of Congregationalism.

In 1957, Congregational Churches which were members of "The General Council of Congregational Christian churches in the United States" merged with the "Evangelical and Reformed Church" to form "The United Church of Christ".

So, today the Puritan churches have either remained "Congregational Churches" or 'morphed' into "United Church of Christ Churches", or 'morphed' into "Unitarian Churches" in the main. (There have also been some smaller and less important 'morphings'.)