Is it obligatory to compensate someone whose shoe fell unintentionally in a place difficult to reach, given that searching for it or for the temporary Imam of the mosque is not possible?

You should have waited for the man and informed him of what happened. Since you did not do so, you cannot be held liable for the value of the shoe, because the default is the exoneration of liability. However, if you wish to donate its value on his behalf, that would be good.