  This paper pursues positive characteristic analogues of the results of
Furusho, Komori, Matsumoto and Tsumura on $p$-adic multiple $L$-functions. We
consider $\infty$-adic and $v$-adic multiple zeta functions concerned by
Angl\`{e}s, Ngo Dac and Tavares Ribeiro. Our main results in this paper consist
of: (1) integral expressions of special values of $\infty$-adic multiple zeta
functions, (2) integral expression of $v$-adic multiple zeta functions
themselves similar to those of $p$-adic multiple $L$-functions, (3) Kummer-type
congruence for the special values of $v$-adic multiple zeta functions at
integers (4) relationships between special values of $\infty$-adic and those of
$v$-adic multipe zeta functions at negative integers and (5) orthognal
properties of multiple zeta functions and multiple zeta star functions.
