Two PDFs (or CDFs) the same if "x" values different

I've been given two sets of data and I need to figure out if they're the "same". Each data set consists of a PDF and a CDF (but not the underlying samples). The PDF and CDF data is in the form of x, y values and they're digitized samples of continuous data. I don't know what the underlying distribution is (I don't know if the underlying distribution is beta, or log-normal etc.) I have data that looks something like:

Set1:

• x1: [1023, 1024, 1025, ...]
• CDF1: [0, 0.001, 0.0025, ....]
• PDF1: [0, 0.001, 0.0015, ...]
• Length1: n1

Set2:

• x2: [1021.5, 1026.5, 1031.5, ...]
• CDF2: [0, 0.002, 0.0043, ....]
• PDF2: [0, 0.002, 0.0023 ...]
• Length2: n2

Importantly:

• n1 != n2 - the lengths are not the same, meaning the data sets don't "line up".
• Sometimes, x1[0] != x2[0] - the first x values will mostly be the same, but sometimes they'll be slightly different
• Sometimes, x1[-1] != x2[-1] - the final x values will mostly be the same, but sometimes they'll be slightly different

Obviously, I thought of a Kolmogorov–Smirnov test, but my reading suggests the test depends on the CDF values 'lining up' on the x axis, which is not my case here. I thought of interpolating my data sets to the same x-axis, but I'm concerned that will introduce fitting errors.

I looked at other non-parametric tests, but it seems to be the same problem everywhere because the x-values don't align.

Are there non-parametric tests that don't rely on CDF/PDF data sampled at the same interval?

I followed the hints from Robert Dodier and did not use KS. I used a polygon representation of both data sets and calculated the area difference. If the difference was greater than a threshold, the test failed. The K-S test was the wrong approach here. Interpolating the results to the same "grid" introduced numerical artifacts that confused the results. Morale of the story: tests have limits.