Why didn't Felix, Festus, and Agrippa free Paul?

Paul was in the custody of Claudius Lysias, the commander of the Roman barracks in Jerusalem, Acts 23:

11 The following night the Lord stood by him [Paul] and said, “Take courage, for as you have testified to the facts about me in Jerusalem, so you must testify also in Rome.”

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God arranged a way for Paul to testify in Rome through the Romans.

The commander learned of a plot to kill Paul. For Paul's safety, he decided to hand over Paul's custody to governor Felix in Caesarea.

Felix held a formal trial and talked to Paul privately many times but did not free him, Acts 24:

26b he hoped that money would be given him by Paul. So he sent for him often and conversed with him. 27 When two years had elapsed, Felix was succeeded by Porcius Festus. And desiring to do the Jews a favor, Felix left Paul in prison.

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Festus held a trial. In the end, Acts 25:

9 But Festus, wishing to do the Jews a favor, said to Paul, “Do you wish to go up to Jerusalem and there be tried on these charges before me?”
10 But Paul said, “I am standing before Caesar’s tribunal, where I ought to be tried. To the Jews I have done no wrong, as you yourself know very well. 11 If then I am a wrongdoer and have committed anything for which I deserve to die, I do not seek to escape death. But if there is nothing to their charges against me, no one can give me up to them [Jewish authority in Jerusalem]. I appeal to Caesar.”

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Some Jews had plotted to kill him. Paul sensed that they would try again. He figured that the safer place for him was under the protection of the Roman authority.

There was yet another trial/hearing:

13 Now when some days had passed, Agrippa the king and Bernice arrived at Caesarea and greeted Festus.

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Festus was interested in Agrippa's opinion on the case, he said to him:

27 "it seems to me unreasonable, in sending a prisoner, not to indicate the charges against him.”

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That was the purpose of the hearing from Festus' point of view.

King Agrippa was also interested in the case and held a hearing. In the end, Acts 26:

30 the king rose, and the governor and Bernice and those who were sitting with them. 31 And when they had withdrawn, they said to one another, “This man is doing nothing to deserve death or imprisonment.” 32 And Agrippa said to Festus, “This man could have been set free if he had not appealed to Caesar.”

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Why didn't Agrippa free Paul?

It wasn't Agrippa's jurisdiction/authority. His input was only advisory to Festus.

Why didn't the Roman authorities free Paul?

Horizontally, Paul explained in Acts 28:

18 When they [Romans] had examined me, they wished to set me at liberty, because there was no reason for the death penalty in my case. 19 But because the Jews objected, I was compelled to appeal to Caesar—though I had no charge to bring against my nation.

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Felix and Festus wanted to do the Jews a favor.

Vertically, Paul explained in Acts 27:

23 For this very night there stood before me an angel of the God to whom I belong and whom I worship, 24a and he said, ‘Do not be afraid, Paul; you must stand before Caesar.'

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Not only Paul must testify in Rome (Acts 23:11), but also he must stand before Caesar.