This is a very commonly encountered problem when working with multi-source spatial data. You might want to consider an important side effect of scaling. I am guessing that after scaling the data, you are planning to sample from it, probably all pixels or a subset of pixels. Your choice of resolution will affect the number of instances that you can sample from. If you scale everything to 4000-meter resolution, you will have larger and less no. of pixels to sample from. Similarly, if you scale everything to 500-meter resolution, you will have smaller and more no. of pixels (64 times more pixels compared to 4000-meter resolution) to sample from.

Another important consideration, changing the resolution of images only either adds pixels (when we upscale and increase resolution) or removes pixels (when we downscale and decrease resolution).

• Upscaling usually takes a large pixel and breaks it down to smaller pixels assigning the same value to the new pixels
• Downscaling usually takes multiple smaller pixels and aggregates them by some method (summation, mean etc.) to form a larger pixel.

More information on scaling: https://developers.google.com/earth-engine/guides/scale

As you can see that while neither of these methods 'add new information' , upscaling preserves the information, whereas if you use downscaling that you might lose spatial granularity of data.

However, when you consider the fact that the multi-source data serves different functions (one is a feature/explanatory variable and another is a target/response variable), we have another issue to consider. Your label resolution is higher than feature resolution. If you were to upscale your explanatory variable to 500-meter resolution, it would just break the large pixels into smaller ones and assign all of them with the same value, whereas each of these pixels could possible have different targets (because response variable is 500-meter resolution). This implies that in your processed data, same value of explanatory variable would lead to multiple different values of response variables. This violates the basic property of functions; one-to-many relations are usually not modeled as functions. For your case, the best option might be to downscale the response variable to 4000-meters resolution. Of course, this would lead to information loss when you aggregate the response variable data. I think this is the most scientific and logical approach in my opinion.