On our site we have previously favoured the view of the Hanafis and Maalikis, that it is permissible to do tayammum using anything on the face of the earth, such as stones. Please see the answer to question number 36774.
The way in which tayammum is done with stones is by striking it once with the palms, then wiping the entire face, then wiping the hands.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
The way in which tayammum is done is by striking clean earth with the hands once, then wiping the entire face with them, then wiping the hands with one another. End quote.
Majmoo ‘ al-Fataawa, 11/155
Shaykh Ibn Baaz (may Allah have mercy on him) said:
He should mention the name of Allah by saying Bismillah, as he does when using water for wudoo'. Then he should strike the dust with them and wipe his face and hands, then say: “Ashhadu an laa ilaaha illa-Allaah wahdahu laa shareeka lah, wa ash-hadu anna Muhammadan ‘abduhu wa rasooluh. Allaahumma-j’alni min al-tawwaabeen waj’alni min al-mutatahhireen (I bear witness that there is no god but Allaah Alone, with no partner or associate, and I bear witness that Muhammad is His slave and Messenger. O Allaah, make me one of those who repent and make me one of those who are purify themselves), as he does when using water, because this takes the place of doing wudoo' with water. End quote.
Majmoo‘ Fataawa Ibn Baaz, 29/100
If the stone is small, there is nothing wrong with turning it over in one's hands like soap, and this turning it over is an alternative to striking it with your hand. After turning it over, one should wipe the face and then the hands.
However, it is better to do tayammum with dust, if possible, because that is more on the safe side and thus the tayammum will be correct according to all the scholars. Some of the Maaliki scholars stated that in Mawaahib al-Jaleel, 1/351.
And Allah knows best.