So after some further research, I've come up with a solution by myself. I will try to explain the procedure here.

Create your base image
Take all your photos and stack them in software such as Sequator or DeepSkyStacker. When using DSS, you will have to create two stacks - one for the stars and one for the foreground - and blend them together in software such as Photoshop.
The meteors won't be visible in thie base image.

Find images with Meteors
Browse through all your images (e.g. in Adobe Bridge or Lightroom - but your standard photo viewing app should also be fine if it supports RAW images). Somehow mark the images with meteors that you would like to include in your final composite.
Open the meteor images and your base image in Photoshop (or other software, like Gimp).

Aligning the base and meteor photos
You will have to align the meteor images to your base image (if you used a star tracker, they already might be aligned). Do so by setting the blending mode of your meteor images to Lighten - this will only show the stars and meteors.
It is best if you have your celestial pole in frame - in the northern hemisphere, this is Polaris. For each meteor image layer, enter transform mode (Crtl+T) and set the anchor point to Polaris (If you cannot see it, Alt+Click). Rotate the image until it properly aligns with the base image (due to lens distortion, we will not get a perfectly aligned image. But this is not too important).
If the celestial pole is not in the frame, it is a bit harder to align the images. You will have to do some guesswork and it will take more time.

Blending the images
Once all the meteor images are aligned to the ground layer, hide all meteor images except for one. Create a negative layer mask (Alt+Click on the layer mask button; this will create a layer mask filled with black) and select the brush tool (you could also use the pen tool). Then, paint white (on the layer mask) over the area where the meteor(s) are visible in this image. You should take some time and make sure that only the meteor is selected and no additional stars. Do this for every meteor image.

You can now do some further post-processing.

The process of blending meteor images is also explained in this tutorial - the only difference is that there is no base images stacked for noise reduction.