Background: As a widely adopted coefficient, Cohen's $\kappa$ is highly affected by marginal probabilities and population composition to measure the degree of agreement. Gwet's $AC_1$ overcomes these limitations and becomes a stable index with the better statistic properties. Stratified analysis is an effective approach to adjust the bias caused by subject covariance in practice. Homogeneity score and goodness-of-fit tests for $AC_1$ have been developed in $K$ independent strata. However, these two tests poorly performed because of the conservative type I error rates (TIEs) in relatively small sample sizes. Exact methods can improve the effectiveness of tests in small samples, and the relevant literature is less. This article proposes several asymptotic and exact methods for homogeneity test of stratified $AC_1$'s under large and small sample sizes. Methods: Three asymptotic test statistics are derived for large sample sizes, including likelihood ratio, score and wald-type tests. Based on these statistics, three exact E, M and E+M approaches are proposed at small sample scenarios. Numerical studies are conducted to verify the effectiveness of aforementioned methods. Two examples are given to illustrate their performance. Results: In large sample scenarios, the TIEs of likelihood ratio and score tests are close to the given nominal level. Likelihood ratio test becomes} more robust as the stratum number increases. Aiming at small sample scenarios, E approach under likelihood ratio and score statistics have better performance for any stratum number. Conclusions: The likelihood ratio statistic is recommended for the homogeneity test of stratified $AC_1$'s in large sample sizes. Exact methods can effectively improve the conservative performance of TIEs in small sample sizes, in which E approach under likelihood ratio and score statistics are more robust.