Mishna Shabbat 7, 3:
THEY ALSO STATED ANOTHER GENERAL PRINCIPLE: WHATEVER IS FIT TO PUT AWAY AND SUCH IS [GENERALLY] PUT AWAY, AND ONE CARRIES IT OUT ON THE SABBATH, HE IS LIABLE TO A SIN-OFFERING ON ITS ACCOUNT.
BUT WHATEVER IS NOT FIT TO PUT AWAY AND SUCH IS NOT [GENERALLY] PUT AWAY, AND ONE CARRIES IT OUT ON THE SABBATH, ONLY HE THAT PUT IT AWAY IS LIABLE.
Mishna Shabbat 9, 6, about specific dusts:
[PIECES] OF THE ALTAR STONES OR THE ALTAR EARTH, MOTH-EATEN SCROLLS OR THEIR MOTH-EATEN MANTLES, OF WHATEVER QUANTITY, [HE IS CULPABLE].
Mishna Eruvin 10, 5:
R`JUDAH RULED: EVEN WHERE A PERSON'S SPITTLE ACCUMULATED IN HIS MOUTH, HE MUST NOT WALK FOUR CUBITS BEFORE HE SPAT OUT.
See SA OC 350, 3 and BetYosef there.
Drool that he want to spite in the mouth is a pb of hotsaa according to rabbi Yehuda. Poskim discusse if this is the halacha. The prohibition is perhaps deorayta (see Beur Halacha siman 350)
For dust see SA hilchot Shabbat 301, 46
to walk with a dress soaked is prohibited rabbinically because of th risk to spin it. But hotsaa is not a problem, despite that the water is not a part of the dress.
Conclusion:
The question about nasal mucus which was moved and come down from is initial place is a problem, may be even deorayita (Beur Halacha 350), according to the mishna in Eruvin. As if it is somewhat you want to launch. To launch is prohibited deorayta because the action of launching give importance to the launched object. Carrying an object from one place to another is mel'echet hotsaa if this object has an effective/potential value, function. This is not the case for dust.
