From: Rabbi Akiva Eiger
To: Rabbi Moshe
Rabbi Akiva Eiger rules that carrying an infant in a karmelis is forbidden, defending the Magen Avraham against the Pri Tevuah's lenient position. For a very young child who cannot yet walk, the prohibition is unambiguous, as the Ran and Rashba classify such a child like an animal for carrying purposes. Regarding the general public, he invokes mutav she'yihyu shoggin, but instructs that scrupulous individuals who ask must be ruled strictly, and that carrying an infant to a bris milah is certainly forbidden even in a karmelis.
Source: Shu"t Rabbi Akiva Eiger §28