Recently, a lot of concern has been raised about assumptions needed in order to fit statistical models to incomplete multivariate and longitudinal data. In response, research efforts are being devoted to the development of tools that assess the sensitivity of such models to often strong but always, at least in part, unverifiable assumptions. Many efforts have been devoted to longitudinal data, primarily in the selection model context, although some researchers have expressed interest in the pattern-mixture setting as well. A promising tool, proposed by Verbeke et al. (2001, Biometrics 57, 43-50), is based on local influence (Cook, 1986, Journal of the Royal Statistical Society, Series B 48, 133-169). These authors considered the Diggle and Kenward (1994, Applied Statistics 43, 49-93) model, which is based on a selection model, integrating a linear mixed model for continuous outcomes with logistic regression for dropout. In this article, we show that a similar idea can be developed for multivariate and longitudinal binary data, subject to nonmonotone missingness. We focus on the model proposed by Baker, Rosenberger, and DerSimonian (1992, Statistics in Medicine 11, 643-657). The original model is first extended to allow for (possibly continuous) covariates, whereafter a local influence strategy is developed to support the model-building process. The model is able to deal with nonmonotone missingness but has some limitations as well, stemming from the conditional nature of the model parameters. Some analytical insight is provided into the behavior of the local influence graphs.