  We study the behavior of the Iwasawa invariants of the Iwasawa modules which
appear in Kato's main conjecture without $p$-adic $L$-functions under
congruences. It generalizes the work of Greenberg-Vatsal,
Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us
simultaneously. As a consequence, we establish the propagation of Kato's main
conjecture for modular forms of higher weight at arbitrary good prime under the
assumption on the mod $p$ non-vanishing of Kato's zeta elements. The
application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is
also discussed.
