Importing content from EE with FeedMe leaves some data out
So the answer was that each of my blocks needed to be wrapped by another element, e.g.
&lt;modularContent&gt;
  &lt;block&gt;&lt;blockHeading&gt;&lt;![CDATA[News 1]]&gt;&lt;/blockHeading&gt;&lt;/block&gt;
  &lt;block&gt;&lt;blockContent&gt;&lt;![CDATA[&lt;p&gt;Lorem ipsum dolor sit amet, consectetur adipisicing elit. Iure rerum in tempore sit ducimus doloribus quod commodi eligendi ipsam porro non fugiat nisi eaque delectus harum aspernatur recusandae incidunt quasi.&lt;/p&gt;]]&gt;&lt;/blockContent&gt;&lt;/block&gt;
&lt;/modularContent&gt;