Pyth, 16 bytes

u|pH<GJxGH>GJwz

Try it online: Demonstration

This is quite hacky. Stack-based languages might have an advantage here.

Explanation

                   implicit: z = 1st input line, w = 2nd
u             wz   reduce, start with G = z
                   for each H in w, update G to:
        xGH          index of H in G
       h             +1
      J              store in J
    <GJ              substring: G[:J] (everything before index J)
  pH                 print substring than H (without newlines)
 |                   afterwards (actually or, but p always returns 0)
           >GJ       substring: G[J:] (everything from index J to end)
                     update G with ^
                   afterwards implicitly print the remainder G