Slowly turn a string into another

V, 14 bytes

òYp$xhòjòÄ$xhh

Try it online!

Explanation:

ò     ò     "Recursively:
 Yp         "  Yank the current line and paste it
   $        "  Move to the end of the current line
    x       "  Delete one character
     h      "  Move One character to the right.
            "  Because of the way loops work in V, this will throw an error if there
            "  Is only one character on the current line.

Now, the buffer looks like this:

0123456789
012345678
01234567
0123456
012345
01234
0123
012
01
0

We just need to do the same thing in reverse for the next line:

j           "Move down one line
 ò     ò    "Recursively (The second ò is implicit)
  Ä         "  Duplicate this line up
   $        "  Move to the end of the current line
    x       "  Delete one character
     hh     "  Move two characters to the right.
            "  Because of the way loops work in V, this will throw an error if there
            "  Is only two characters on the current line.

More interesting alternate solution:

òÄ$xhòç^/:m0
ddGp@qd

Pyth, 9 bytes

j+_._Et._

A program that takes the second string, and then the first string, as quoted strings on STDIN and prints the result.

Try it online

How it works

j+_._Et._  Program. Inputs: Q, E
   ._E     Yield prefixes of E as a list
  _        Reverse the above
       ._  Yield prefixes of Q as a list (implicit input fill)
      t    All but the first element of above
 +         Merge the two lists
j          Join on newlines
           Implicitly print

Python, 93 bytes

f=lambda a,b,r='',i=2:a and f(a[:-1],b,r+a+'\n')or(len(b)>=i and f(a,b,r+b[:i]+'\n',i+1)or r)

Starts with the empty string r, adds a and a newline and removes the last character from a until a is empty then adds the required portions of b and a newline by keeping a counter, i, which starts at 2 until the length of b is exceeded, then returns r. Has a trailing newline.

All tests are on ideone