This Halacha is an excerpt from our Sefer
Rinsing the mouth on a fast day:
On a public fast day, one is not to rinse his mouth in the morning. This applies even if one will be using less than a Revius of water. [If, however, this is causing one discomfort, he may be lenient to rinse his mouth.] On a private fast day, it is permitted to rinse the mouth with water, even if one uses more than a Revius. It is however forbidden to rinse the mouth on a private fast day with other liquids if they contain more than a Revius. If however the liquids are not drinkable, such as vinegar, then it is permitted to use even more than a Revius [on a private fast day].
May one brush his teeth on a fast day?
If not doing so will cause one pain or discomfort it is allowed.
May one use mouthwash if not doing so will cause pain and discomfort?
Yes, it may be used in such a case.
 Michaber 567:3 “One who is accustomed to rinse his mouth, it is improper-not Kosher- to do so on a public fast day” based on Terumos Hadeshen 158 regarding the prohibition to taste foods on a fast; Mamar Mordechai 567:2; Levush 567; Nehar Shalom 567:1; Bigdei Yesha 567:6; Kaf Hachaim 567:13; Chayeh Adam 132:20; Ketzos Hashulchan 2 footnote 25 in name of Ateres Zikeinim; Sefer Haminhagim p. 4 [English]
Other opinions: Some Poskim rule it is permitted to rinse one’s mouth in water even on a public fast day, with exception to Tisha B’av and Yom Kippur. [Drashos Maharil, brought in Darkei Moshe 567:1 and Elya Raba 567:5 and Kaf Hachaim 567:14] The Elya Raba ibid concludes one may be lenient if he is accustomed to do so daily.
Stringency or prohibition? It is implied from the Michaber ibid that there is no prohibition involved in rinsing the mouth even on Yom Kippur being one has no intent to swallow the liquid and it is merely water. [P”M 567 A”A 6]
 The reason: As one may come to swallow the water. [Mamar Mordechai 567:2; Levush 567, brought in P”M ibid]
 Machatzis Hashekel 567:6; Nehar Shalom 567:1; Bigdei Yesha 567:6; Kaf Hachaim 567:13
Other opinions: Some Poskim rule it is permitted to rinse one’s mouth with less than a Revius of water, as the entire prohibition against tasting foods only applies on Yom Kippur and Tisha B’av. [M”A 567:6; See P”M 567 A”A 6 who questions his ruling; See also Machatzis Hashekel ibid who learns in M”A that only on a Tannis Tzibur that is not one of the four fasts is this allowed, however on the four fasts even less than a Revius is forbidden.]
 Chayeh Adam 132:20; M”B 567:11; Kaf Hachaim 567:14
 Michaber ibid; Terumos Hadeshen ibid
 M”A 567:7 based on Terumos Hadeshen ibid
 M”A ibid; Elya Raba 567:4; M”B 567:12; It is unclear if the M”A is referring to only a private fast day or even a public fast day. However seemingly he is referring to a private fast day, which is the discussion of M”A ibid, and so is implied from M”B ibid and so writes M”E 602 Alef Lamateh 3; Piskeiy Teshuvos 567 footnote 2.
 Minchas Yitzchak 4:109; Piskeiy Teshuvos 567:1; based on Michaber 567:3 and M”B 11 who allows to rinse ones mouth in a case of pain
 See M”A 567:3 that one may rinse his mouth using vinegar, being it is not fit for drinking [accordingly, it is allowed even not in a time of need]; Kinyan Torah 2:49; See Mateh Efraim 602:3; Piskeiy Teshuvos 567 footnote 3