The Mishnah in Taanis 2 (1) refers to the practice of taking the Ark out of the synagogue into the the town square on the more serious fasts for rain and putting ashes on the ark:
סדר תענייות כיצד:: מוציאין את התיבה לרחובה של עיר; ונותנין אפר מקלה על גבי התיבה
Modern arks come in at least two forms. One is a movable ark and the other is built into the structure of the synagogue.
I do not know of a widespread custom to take the Ark out of the synagogue into the the town square any more. Would it nevertheless be preferable to have a movable ark to remain in touch with the practice described in the Mishnah?