Sequence-processing neural networks led to remarkable progress on many NLP tasks. As a consequence, there has been increasing interest in understanding to what extent they process language as humans do. We aim here to uncover which biases such models display with respect to "natural" word-order constraints. We train models to communicate about paths in a simple gridworld, using miniature languages that reflect or violate various natural language trends, such as the tendency to avoid redundancy or to minimize long-distance dependencies. We study how the controlled characteristics of our miniature languages affect individual learning and their stability across multiple network generations. The results draw a mixed picture. On the one hand, neural networks show a strong tendency to avoid long-distance dependencies. On the other hand, there is no clear preference for the efficient, non-redundant encoding of information that is widely attested in natural language. We thus suggest inoculating a notion of "effort" into neural networks, as a possible way to make their linguistic behavior more human-like.