The are some ways you may move the stone on Shabbat.
To set it aside before Shabbat to be used on Shabbat.
To use the stone once before Shabbat, keep the door open.
To kick the stone in place.
The Gemara in Shabbat 50a: Rabbah bar bar Chana taught palm leaves which one put side for firewood and he decided to sit on it he needs to tie up the leaves. Rashba”g said you do not need to tie it up. He taught it and said the halacha is like Rashba’g. Rav said you ne to tie it and Shmuel said you just need to think about them for sitting. Rav Asi said he just needs to sit on it even though you did not tie it or think about it.
The Rishonim argue who we rule like.
Rabbeinu Tam ruled like Rav that one needs to tie it up before Shabbat and to designate them for sitting on Shabbat. Rabbeinu Shimshon argues and rules like Rav Asi that you just need to sit on it before Shabbat without designating it.
The Shulchan Aruch rules like Rabbeinu Shimshon(OH308:20): Palm leaves which were cut down to be burnt are Muktzeh and they are forbidden to be moved. If you sat on them a bit before Shabbat it is permitted on sit on them on Shabbat. Especially if you tied them up like a mat to sit on or designated them for that purpose during the week.
Similarly, the Gemara in Shabbat 125b quotes R’ Chanina: Onetime Rebbi went somewhere and found a pile of rocks (Rashi set up for building). He said to his students go designate them so I may sit on them on Shabbat and Rebbi did not require them to do any action. R’ Yochanan said Rebbi did require them to do something. What was it? Reb tole theme to set them up for sitting so he would not need to touch it the on Shabbat. R’ Assi said he told them to dust them off a bit. (Rash I a minor action.)
R’ Chanina and R’ Yochanan argue what to do to make building stones sittable on Shabbat. R’ Chanina said just like by the palm leaves it is enough with just designating them to sit on. R’ Yohanan held that stones are even more stringent then palm leaves and they need action done to them.
The Rambam (Shabbat 25:21) writes : A pile of rocks that one designated for use before Shabbat, if he set them up to sit on fine, if not then it is forbidden to is on them.
The Rambam is of the opinion that a pile of rocks is more stringent them palm leaves. The Maggid Mishna explains the difference that palm leaves can be either for fire or for mats, therefore even if you though to make it into firewood you can change your mind. Stones are themselves not made for sitting so you need to do more than change you mind you need to make them viable for sitting.
The Rif does not even mention the Sugya of stones and he understands it is the same as palm leaves, and it is enough to just designate them before Shabbat.
The Shulchan Aruch and the Rema argue who to Posek like. The Shulchan Aruch writes: A pile of stones however even though we designated them before Shabbat for sitting you may not sit on them unless you arrange them. The Rema argues that they have the same halacha as Palm leaves.
If seems that it is not enough to just think about them. the Gemara in Shabbat though on daf 50a sys: Rav Yehuda said one may fill his container with dirt and use it. Mar Zutra in the name of Mar Zutra Rabbah said you must designate a corner for it.
It seems in order to designate something for Shabbat you need to also designate a special place to keep it. So rules the Shulchan Aruch 308:38. It seems that in ordo t designate something for use on Shabbat you just use the object for something belter then what is it now. Therefore, the Rashba writes that one may not designate a rock to use a rock on Shabbat. The Ran comments that you may designate a rock for sue a as a rock on Shabbat that is only if it is for one Shabbat. However, if you want to permanently change its status then he has to say so.
The Shibolei Haleket that either way you need to designate forever. The Mordechai adds the one may also to do action before Shabbat.
The Shulchan Aruch rules like the Rashba the designating of a rock as a rock can only be for one Shabbat not for permanent Shabbat use. In the case of the doorstop it is normal use and just designating it ne Shabbat is enough. Even just using as such during the week will work as well and it will be permitted to move it. So writes the Mishna Berura and Yalkut Yosef.