The Drupal's core update.php seems not to update URLs in the Flexinode's fields to the new form starting with slash, as it does for ordinary nodes' content (e.g., update #178).
As a workaround one may use the following SQL commands to update links in the text fields:
UPDATE flexinode_data SET textual_data=replace(textual_data,'href=\"node','href=\"/node');
UPDATE flexinode_data SET textual_data=replace(textual_data,'href=\"files','href=\"/files');
UPDATE flexinode_data SET textual_data=replace(textual_data,'src=\"files','src=\"/files');