Substitute a string with itself!

CJam, 11 bytes

q__1m>.{er}

Test it here.

Explanation

q__    e# Read input and make two copies.
1m>    e# Rotate the second copy one character to the right.
.{er}  e# For each pair of characters from the second and third string,
       e# replace occurrences of the first with the second.

TeaScript, 17 bytes 19 21 24

TeaScript is JavaScript for golfing

xd(#lg(i,xC(1#a))

Nice and short

Try it online (watch for trailing whitespace in the input)

Ungolfed & Explanation

x.reduce     // Reduce over input
  (#         // Anonymous function expands to ((l,i,a)=>
    l.g(        // global replace...
     i          // replace var i with...
     x.cycle(1) // Cycle x 1
     [a]        // At position a
    )
  )

JavaScript ES6, 69 bytes

s=>[...s].reduce((p,c,i)=>p.replace(RegExp(c,'g'),s.slice(i-1)[0]),s)

F=s=>[...s].reduce((p,c,i)=>p.replace(RegExp(c,'g'),s.slice(i-1)[0]),s)

input.oninput=()=>output.innerHTML=F(input.value)

#input, #output {
  width: 100%;
}

<textarea id='input' rows="5">
</textarea>
<div id="output"></div>