Two key ingredients to carry out inference on the copula of multivariate observations are the empirical copula process and an appropriate resampling scheme for the latter. Among the existing techniques used for i.i.d. observations, the multiplier bootstrap of Rémillard and Scaillet (J. Multivariate Anal. 100 (2009) 377-386) frequently appears to lead to inference procedures with the best finite-sample properties. Bücher and Ruppert (J. Multivariate Anal. 116 (2013) 208-229) recently proposed an extension of this technique to strictly stationary strongly mixing observations by adapting the dependent multiplier bootstrap of Bühlmann (The blockwise bootstrap in time series and empirical processes (1993) ETH Zürich, Section 3.3) to the empirical copula process. The main contribution of this work is a generalization of the multiplier resampling scheme proposed by Bücher and Ruppert along two directions. First, the resampling scheme is now genuinely sequential, thereby allowing to transpose to the strongly mixing setting many of the existing multiplier tests on the unknown copula, including nonparametric tests for change-point detection. Second, the resampling scheme is now fully automatic as a data-adaptive procedure is proposed which can be used to estimate the bandwidth parameter. A simulation study is used to investigate the finite-sample performance of the resampling scheme and provides suggestions on how to choose several additional parameters. As by-products of this work, the validity of a sequential version of the dependent multiplier bootstrap for empirical processes of Bühlmann is obtained under weaker conditions on the strong mixing coefficients and the multipliers, and the weak convergence of the sequential empirical copula process is established under many serial dependence conditions.