My answer is that "it is not possible to say anything about this in general". For example,
1. Given a sample from the normal distribution, its variance of sample median is about 1.57 times the variance of the sample mean.
2. Given a sample from the Cauchy distribution, its sample mean does not mean anything because the distribution mean does not exist at all. But you can use the sample median to estimate the distribution median, and the variance of the sample median also exists.

The variance of an iid-sample median equals 1/(4nf(m)*f(m)), where f() is the probability density function of the relevant response, m is the true response median, n is the size of the sample. (iid means independent and identically distributed.)