  The $\gamma$-FDP and $k$-FWER multiple testing error metrics, which are tail
probabilities of the respective error statistics, have become popular recently
as less-stringent alternatives to the FDR and FWER. We propose general and
flexible stepup and stepdown procedures for testing multiple hypotheses about
sequential (or streaming) data that simultaneously control both the type I and
II versions of $\gamma$-FDP, or $k$-FWER. The error control holds regardless of
the dependence between data streams, which may be of arbitrary size and shape.
All that is needed is a test statistic for each data stream that controls the
conventional type I and II error probabilities, and no information or
assumptions are required about the joint distribution of the statistics or data
streams. The procedures can be used with sequential, group sequential,
truncated, or other sampling schemes. We give recommendations for the
procedures' implementation including closed-form expressions for the needed
critical values in some commonly-encountered testing situations. The proposed
sequential procedures are compared with each other and with comparable fixed
sample size procedures in the context of strongly positively correlated
Gaussian data streams. For this setting we conclude that both the stepup and
stepdown sequential procedures provide substantial savings over the fixed
sample procedures in terms of expected sample size, and the stepup procedure
performs slightly but consistently better than the stepdown for $\gamma$-FDP
control, with the relationship reversed for $k$-FWER control.
