The Metzitzah process:[1]

Immediately after the Peria is done, the Mohel is to quickly perform the Metzitza.[2] The Metzitza is to be performed by the Mohel sucking the Mila until blood comes out from all the inner areas of the limb.[3] [By a circumcision [sucking the blood of the circumcised area] was only permitted because [lack of doing so] poses danger.] Any Mohel who refuses to perform the Metzitza is to be fired.[4]

With the mouth:[5] The Metzitza is an integral part of the Mitzvah of circumcision and must be performed specifically with the mouth. It may not be performed using a tube. A Mohel who refuses to perform the Metzitza with his mouth and desires to use a tube to do so, is not to be allowed to circumcise.[6] [Nonetheless, if it is not possible to perform the Metzitza with his mouth, such as due to refusal of the parents to permit a circumcision to take place with it, then it is better for the Metzitza to be performed using a tube then for it to not be performed at all. In such an event, one is to use a glass tube which contains a small ball of cotton that has a disinfection agent absorbed in it.[7]]

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[1] Michaber Y.D. 264:3; 328:53; Admur 328:54; 331:1; Shabbos 133b; Shevach Baris 18:6; Pesakim Uteshuvos 264:14-15; See Encyclopedia Talmudit Vol. 46 Erech Mila p. 472-476

[2] Siddur Admur; Michaber Y.D. 264:3;  Kuntrus Hasiddur of Gra”ch Na”ah 11; Shevach Baris 18:6

[3] Michaber Y.D. 264:3; Admur 328:54; 331:1; Shabbos 133b

The reason for Metzitza: Metzitza must be performed in order to suck out the blood from the distanced areas of the limb in order to prevent danger to the child. [Michaber Y.D. 264:3; Admur 328:54; 331:1; Shabbos 133b] See also Pesakim Uteshuvos 264:13 footnotes 108-111

[4] Michaber ibid

[5] See Binyan Tziyon 23;24; Maharam Shick Y.D. 244; Mahariy Asad Y.D. 258; Tzafnas Paneiach 2:152; Sdei Chemed Kuntrus Hametzitza; Igros Kodesh Rashab 1:387; Igros Kodesh 3:176; 15:92; Shulchan Menachem 5:132; Shevach Baris 18:6 ; Pesakim Uteshuvos 264:15 in length

No known health risk: Experience from the past thousands of years shows that there is no worry of danger involved in performing the Metzitza directly with one’s mouth. [Igros Kodesh Rashab 1:387 based on Meishiv Nefesh; Maharam Shick ibid]

[6] Igros Kodesh Rashab 1:387

[7] Igros Kodesh 15:341; Kefar Chabad vol. 708 p. 166; ; Sdei Chemed; Chasam Sofer; See Pesakim Uteshuvos 264:15 footnotes 122-124