We propose two coding schemes for the two-receiver discrete memoryless broadcast channel (BC) with rate-limited feedback from one or both receivers. They improve over the nofeedback capacity region for a large class of channels, including the class of \emph{strictly essentially less-noisy BCs} that we introduce in this article. Examples of strictly essentially less-noisy BCs are the binary symmetric BC (BSBC) or the binary erasure BC (BEBC) with unequal cross-over or erasure probabilities at the two receivers. When the feedback rates are sufficiently large, our schemes recover all previously known capacity results for discrete memoryless BCs with feedback. In both our schemes, we let the receivers feed back quantization messages about their receive signals. In the first scheme, the transmitter simply \emph{relays} the quantization information obtained from Receiver 1 to Receiver 2, and vice versa. This provides each receiver with a second observation of the input signal and can thus improve its decoding performance unless the BC is physically degraded. Moreover, each receiver uses its knowledge of the quantization message describing its own outputs so as to attain the same performance as if this message had not been transmitted at all. In our second scheme the transmitter first \emph{reconstructs and processes} the quantized output signals, and then sends the outcome as a common update information to both receivers. A special case of our second scheme applies also to memoryless BCs without feedback but with strictly-causal state-information at the transmitter and causal state-information at the receivers. It recovers all previous achievable regions also for this setup with state-information.