This paper derives time-uniform confidence sequences (CS) for causal effects in experimental and observational settings. A confidence sequence for a target parameter is a sequence of confidence intervals such that every one of these intervals simultaneously captures the target with high probability. Such CSs provide valid statistical inference for parameters at arbitrary stopping times, unlike classical fixed-time confidence intervals which require the sample size to be fixed in advance. Existing methods for constructing CSs focus on the nonasymptotic regime where certain assumptions (such as known bounds on the random variables) are imposed, while doubly-robust estimators of causal effects rely on (asymptotic) semiparametric theory. We use sequential versions of central limit theorem arguments to construct large-sample CSs for causal estimands under nonparametric conditions. These CSs allow analysts to update statistical inferences about causal effects in lieu of new data, and studies can be continuously monitored, stopped, or continued for any data-dependent reason, all while controlling the type-I error rate.