Imagine that a variable takes values 0 and 1 with probability both 0.5. Sample from that distribution and most of the medians will be 0 or 1 and a very few exactly 0.5. The means will vary far less. The mean is much more stable in this circumstance.
Here is a sample graph of results. The plots are quantile plots, i.e. ordered values versus plotting position, a modified cumulative probability. The results are for 10,000 bootstrap samples from 1000 values, 500 each 0 and 1. The means range fortuitously but nicely from 0.436 to 0.564 with standard error 0.016. The medians are as said, with standard error 0.493. (Closed-form results are no doubt possible here too, but a graph makes the point vivid for all.)
But that is exceptional. It illustrates the least favourable case for medians, a symmetric bimodal distribution such that the median is likely to flip between different halves of the data. However, symmetric bimodal distributions are not especially common, but watch out for so-called U-shaped distributions in which the extremes are most common and intermediate values uncommon. Distributions that are unimodal, or in which the number of modes has only a small effect on median or mean, are more common.
As advised by every treatment of robust statistics, a very common situation is that your data come with tails heavier than Gaussian and/or with outliers, and in those circumstances median will almost always be more robust. The point is that that is not a universal general result.
All that said, what relevance is a general result? You can at a minimum establish by bootstrapping the relative variability of mean and median for your own data. That's what you care about.