Median:Mean::Percentile:?

Question

Here is a simple question that I am hoping has a simple answer.

The median and mean are both estimates of the central tendency of a random variable. They both have different advantages in different situations, but one advantage of the mean is that it's a continuous function of the probability distribution, in the sense that small adjustments to the probabilities involved lead to small adjustments to the estimate - in particular, this is even true if we have a discrete distribution. For instance, with a Bernoulli-distributed variable, the mean is a continuous function of the parameter $p$ and the median is not.

Now, let's assume that we have a distribution where the mean and median are decently close. Then the mean can be viewed as a "continuous estimator" of the median. So what I want is something like a "continuous estimator" of some other percentile, like the 25th or 5th percentile. Put another way, rather than measuring the central tendency, I want to measure something like an "upper tendency" or a "lower tendency", but do so in a way which is continuous. What are some good ways to do that?

I feel like there ought to be some simple thing which will do. If the median and mean are both estimators of "central tendency", and the quantile function generalizes the median, what similarly generalizes the mean?

Here are some things I've tried:

1. Taking some kind of weighted mean which emphasizes low values. For instance, if we have a one-sided distribution, we can multiply the probability of $r$ by some $1/r^k$ for some k. This can work but be very un-robust as stray noise in the low values can be amplified.
2. Taking the mean, minus some number of standard deviations. This can also work, but we get very different results depending on the kurtosis.
3. If for instance, I want the 12.5th percentile, I can compute that by computing the median, throwing values greater than it away, and repeating two additional times. If we just replace with the mean, we get something analogous, but it isn't continuous.

Is there some "right" way to do this?