A second example is government unemployment insurance payments, which often cover up to some percentage of your previous wage, up to some maximum, and sometimes with a floor. For example, Figure 2, from a recent paper by Landais, shows the unemployment benefit in Louisiana as a function of the highest quarter earnings:

Then the idea is to see whether the outcome of interest also exhibits a kink/change of slope in its relation to the running variable at this same point. For example, Landais is interested in how long people spend in unemployment as function of the amount of benefit they receive. Figure 3 below shows a strong change in the shape of the relationship between unemployment duration and highest quarter earnings at this kink-point.

Figure 3: Kink in the outcome at the same place as kink in the treatment

Then the causal impact is found by dividing the change in slope for the outcome by the change in slope for the treatment.
When the policy rule is implemented with some error, a fuzzy RKD design can be used (see Card et al. (2015))

Estimation
This can be estimated by using a polynomial regression. Let x be the running variable (e.g. highest quarter previous earnings), Y the outcome variable (e.g. duration of unemployment); k the kink point, and D a dummy variable for being to the right of the kink. Then estimate:
Y = a0 + a1*(x-k) + b1*D*(x-k) + a2*(x-k)^2 + a3*D*(x-k)^2 + …
Then b1 tells you the change in the slope of the outcome at the kink point.
Then run the same regression to get the change in the slope of the treatment variable at the kink point of the running variable, and take the ratio of the two coefficients. The standard error can then be bootstrapped or recovered using the Delta method.
Estimation can also be done non-parametrically using local polynomials.

ey assumptions and checks
The key assumptions are similar to those for regression discontinuity and are checked in similar ways:

There should be no change in slope at the kink point for other covariates (e.g. Landais shows no kinks in slope for age, education, capital income, or number of dependents at the kink point)
There is no manipulation of the assignment variable at the kink point – e.g. in the unemployment case, people don’t strategically manipulate their past earnings in expectation of how this will affect their future unemployment benefits. This can be tested by McCrary-type tests that show the distribution of the assignment variable is continuous at the kink point, and also that the first derivative of this pdf is also smooth. Card et al. (2015) show that you can allow agents to do some sorting around the kink points, so long as they make small optimization errors or mistakes so that this sorting is not completely deterministic.
This paper suggests some placebo tests you can do at non-kink points.
Technical details can be found in this Econometrica paper by Card et al. (2015).
Caveats
The RKD design can do pretty poorly in small samples, and typically will require a wider bandwidth or larger sample than you would need with a RDD.
There are methods for optimally choosing the bandwidth, and papers that aren’t able to justify this choice and/or show robustness to different bandwidth choices face criticism. See the comments on this AEJ Economic Policy paper for an example.

Resources:
The AEJ paper of Landais has data and replication Stata code available online.
This Stata Journal article by Calonico et al. has some RD code that also covers the RKD case.