The Shulhan Aruch (Siman 306) permits announcing lost objects on Shabbat, even though the lost item is Mukseh. For example, one may announce that a cell phone was found during the week, and whoever lost it may claim it in the office after Shabbat. Even though this is speech connected to violating Shabbat, it is permitted because it is considered “L’sorech Misva”-for the Misva of returning lost property.
Rav Elyashiv (Jerusalem, 1910-2012) is quoted in the Shemirat Shabbat K’hilhata (Ch. 20, Note 29) as ruling that if one finds his friend’s wallet in the public domain on Shabbat, he is not required to stand guard by the wallet until after Shabbat, even if he would do so for his own wallet. The reason is that the Misva of “Hashavat Aveda”-returning lost property does not become incumbent upon him until he actually picks up the object. There is no obligation to watch a lost object. Of course, if one wants to be a Hasid, he may stand watch.
It is permitted to announce lost property which is Muksa on Shabbat. There is no obligation to stand guard over a lost Muksa item, in order to return it after Shabbat.