Got it. In the preprocess I needed just to put the data in $vars['field_breadcrumb'].

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I have the following line in THEME_preprocess_node

$vars['node']->field_breadcrumb['und'][0]['value'] = theme('breadcrumb', array('breadcrumb'=>$breadcrumb));

In Display Suite, I added a preprocess field. The machine name of the field is field_breadcrumb,

The field appears in the Manage Display settings, but its contents don't appear when the display is rendered.