Ruby, 88 bytes

->s{v=[];while s.length>0;s.reject!(&:empty?);v<<s.map{_1[0]};s.map!{_1[1..]}end;v.join}

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A rather naive approach and my first ruby golf.

• s.reject!(&:empty?) removes all empty strings from the array;
• v<<s.map{_1[0]} the first letters are added to the output vector
• s.map!{_1[1..]} and the first letters are removed
• while s.length>0 the process is repeated until all strings were removed and the output vector is joined to a string