I would do the following:

B[[Permute[Ordering[B[[All,{2,3}]]], Ordering[A[[All,{2,3}]]]]]]

Step 1. for every element in A, there is a corresponding element in B. The first step is to write a function that takes in an element from A and gives the corresponding element from B.

• To do this, first write a function that checks whether an element from A and an element from B match.
• Then use the Select function to make the function in Step 1.

Step 2. Go thru every element in A and replace it with the corresponding element in B.

• The cleanest way to do this is with Map. You may prefer using a For loop or using Replace depending on your coding style.

I'm not sure what your programming experience is. Can you write a function that takes two elements (one from A and one from B) and checks whether they match each other?