Is there a prohibition against throwing a miswak stick on the ground, and is it permissible to break it and throw it in the trash after it dries?

There is no indication of dislike (karaha) or prohibition (tahrim) regarding discarding a miswak stick on the ground, nor are we aware of any scholar holding such an opinion. The miswak was mentioned in a noble hadith (definition) to express insignificance, like the Prophet's (peace be upon him) saying: "Even if it were a stick of Arak." Therefore, there is no impediment to breaking it and disposing of it in the trash.