Python 2, 118 107 103 97 93 bytes

s=''
for i in input()+['']:
 while i.find(s):s=s[:-1];print s
 while i>s:s+=i[len(s)];print s

Pretty elementary solution. Could use some work.

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Revision 3: -6 bytes. Credit goes to @mbomb007 for suggesting to put the whiles on a single line. I also fixed a bug by changing s.find(i) to i.find(s).

Revision 4: -4 bytes. Credit goes to @xnor for realizing that I didn't need to store the input in a variable.

Python 2, 118 107 103 97 93 92 bytes

s=''
for i in input()+[s]:
 while i.find(s):s=s[:-1];print s
 while i>s:s+=i[len(s)];print s

Input is given as ['abc', 'abcdef', 'abcfed'], or as ["abc", "abcdef", "abcfed"].

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Revision 3: -6 bytes. Credit goes to @mbomb007 for suggesting to put the whiles on a single line. I also fixed a bug by changing s.find(i) to i.find(s).

Revision 4: -4 bytes. Credit goes to @xnor for realizing that I didn't need to store the input in a variable.

Revision 5: -1 byte. Credit goes to me for realizing that [''] is the same thing as [s] when adding it to the input.

Python 2, 118 107 103 97 bytes

x=input()+['']
s=''
for i in x:
 while i.find(s):s=s[:-1];print s
 while i>s:s+=i[len(s)];print s

Pretty elementary solution. Could use some work.

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Revision 3: -6 bytes. Credit goes to @mbomb007 for suggesting to put the whiles on a single line. I also fixed a bug by changing s.find(i) to i.find(s).

Python 2, 118 107 103 97 93 bytes

s=''
for i in input()+['']:
 while i.find(s):s=s[:-1];print s
 while i>s:s+=i[len(s)];print s

Pretty elementary solution. Could use some work.

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Revision 3: -6 bytes. Credit goes to @mbomb007 for suggesting to put the whiles on a single line. I also fixed a bug by changing s.find(i) to i.find(s).

Revision 4: -4 bytes. Credit goes to @xnor for realizing that I didn't need to store the input in a variable.

Python 2, 118 107 103 bytes

x=input()+['']
s=''
for i in x:
 while s.find(i):
    s=s[:-1]
    print s
 while i>s:
    s+=i[len(s)]
    print s

Pretty elementary solution. Could use some work.

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Python 2, 118 107 103 97 bytes

x=input()+['']
s=''
for i in x:
 while i.find(s):s=s[:-1];print s
 while i>s:s+=i[len(s)];print s

Pretty elementary solution. Could use some work.

Revision 1: -11 bytes. Credit goes to @xnor for his post on Python golfing tips, and to @Lynn for finding the tip for me, and to me for being smart. Two changes were made: Instead of not s.startswith(i), I used s.find(i), and instead of i!=s I used i>s.

Revision 2: -4 bytes. Credit goes to me realizing I made a really dumb mistake. Instead of using single-tab and double-tab indentation, I used single-space and single-tab indentation.

Revision 3: -6 bytes. Credit goes to @mbomb007 for suggesting to put the whiles on a single line. I also fixed a bug by changing s.find(i) to i.find(s).