Step 1: Read the question.
A case is pending in a civil court. Someone asks to move it from one district to another district within the same State. We must find who can order this transfer.
Step 2: Find the law.
Sections 22, 23 and 24 of the Code of Civil Procedure, 1908 deal with transfer of suits.
Step 3: Understand the purpose.
Transfer powers help ensure a fair trial and the convenience of the parties. A higher court can shift a case when justice needs it.
Step 4: Identify the proper authority.
When the two district courts are under the same High Court but in different districts, the High Court has the power to transfer the case.
Step 5: Check the other options.
Option 1 is wrong because transfer can happen at any stage, not only after trial. Option 2 is wrong because the trial court cannot itself send the case to another district. Option 4 is wrong because the parties need not agree.
Step 6: Final answer.
So an inter-district transfer within a State is ordered by the High Court.
\[ \boxed{\text{By the High Court.}} \]