We focus on the design of distributed Luby transform (DLT) codes for erasure networks with multiple sources and multiple relays, communicating to a single destination. The erasure-floor performance of DLT codes improves with the maximum degree of the relay-degree distribution. However, for conventional DLT codes, the maximum degree is upper-bounded by the number of sources. An additional constraint is that the sources are required to have the same information block length. We introduce a $D$-bit buffer for each source-relay link, which allows the relay to select multiple encoded bits from the same source for the relay-encoding process; thus, the number of sources no longer limits the maximum degree at the relay. Furthermore, the introduction of buffers facilitates the use of different information block sizes across sources. Based on density evolution we develop an asymptotic analytical framework for optimization of the relay-degree distribution. We further integrate techniques for unequal erasure protection into the optimization framework. The proposed codes are considered for both lossless and lossy source-relay links. Numerical examples show that there is no loss in erasure performance for transmission over lossy source-relay links as compared to lossless links. Additional delays, however, may occur. The design framework and our contributions are demonstrated by a number of illustrative examples, showing the improvements obtained by the proposed buffer-based DLT codes.