I'd like to batch convert some log footage as well as transform each clip into rec709 colorspace. Is there any way to apply a custom lut to the footage? If so, what's the correct way to do that?
You may want to use FFMPEG's lut3d filter. It requires you to provide a look-up table (a *.cube file). For example, if you have an ARRI camera you can generate these files using ARRI Color Tool or simply download a package with them from the linked page.
Once you have the files, use FFMPEG like this:
ffmpeg -i "Input.mov" -vf lut3d="ARRIP3D65PQ108-33.cube" -s 1920x1080 -c:v dnxhd -pix_fmt yuv422p -b:v 120M DNxHD_for_Editing.mxf
The important part here is the use of lut3d filter:
where ARRIP3D65PQ108-33.cube is the LUT file to use with the filter.