It is permissible to move the furnishings of an old mosque to another mosque
Yes, it is permissible to do that, in sha Allaah.
Ibn Qudaamah said: If there is anything left over from the mats and adornments of the mosque that is not needed, it is permissible to put them in another mosque, or to give them in charity to poor neighbours and others.
Ahmad said concerning a case where mosque had been built and some wood was left over: Let it be used to help another mosque, or words to that effect.
Al-Marwadhi said: I asked Abu ‘Abd-Allaah about pipes of the mosque, if there is anything left over, or if any wood is left over. He said: Give it in charity. Al-Mughni, 6/219-220.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
Yes, it is permissible to move a waqf if that serves a greater interest. If something is no longer needed in a mosque, such as furnishings or shelves etc, we may move the items themselves to another mosque if possible, otherwise we may sell these things and spend the money earned on the mosque. If it belongs to the Ministry of Awqaaf, then the people in authority are the ones who should decide what is to be done.
Liqa’ al-Baab il-Maftooh, 3/248.
And Allaah knows best.
Although a musalla does not come under exactly the same rulings as the mosque, it is like a mosque to some extent, so moving furnishings to a musalla should take priority over giving them in charity to the poor. And Allaah knows best.