FFmpeg - A/V sync problem when muxing concatenated MP4 files - ffmpeg

Question

First of all, I'd like to say that I've spent the whole day on researching the issue, tried everything I found on the web and none of the solutions worked, so I'm hopeless.
I'm testing live streaming using Azure Media Services. When the event is stopped, I'd like to get the recording as a single MP4 file, but Azure saves the video and audio tracks separately in a fragmented form. I figured out that I can concatenate these files (I used OSX cat command), so I do have perfectly working MP4 files for video and audio - both of them play without any interruptions and seem just fine.
Here's the FFmpeg info on these files:
Input #0, mov,mp4,m4a,3gp,3g2,mj2, from 'video_1200k.mp4':
Metadata:
major_brand : isml
minor_version : 512
compatible_brands: ismlpiff
encoder : Lavf58.45.100
Duration: 01:05:49.57, start: 0.085000, bitrate: 1166 kb/s
Stream #0:0(und): Video: h264 (High) (avc1 / 0x31637661), yuv420p, 1280x720, 1163 kb/s, 23.98 fps, 23.98 tbr, 90k tbn, 47.95 tbc (default)
Metadata:
handler_name : VideoHandler
vendor_id : [0][0][0][0]
Input #1, mov,mp4,m4a,3gp,3g2,mj2, from 'audio_128k.mp4':
Metadata:
major_brand : isml
minor_version : 512
compatible_brands: ismlpiff
encoder : Lavf58.45.100
Duration: 01:06:09.57, start: 0.000000, bitrate: 132 kb/s
Stream #1:0(und): Audio: aac (LC) (mp4a / 0x6134706D), 44100 Hz, stereo, fltp, 128 kb/s (default)
Metadata:
handler_name : Audio
vendor_id : [0][0][0][0]
I noticed the duration difference, but I thought that when I mux these streams, I would get a file in which the last few seconds would have just the audio track. I started with testing the most basic FFmpeg command ffmpeg -i video_1200k.mp4 -i audio_128k.mp4 -c copy muxed.mp4, however the A/V turned out to be out of sync - the video is ahead of audio and the gap increases quite a lot over time.
I've tried pretty much every FFmpeg command arguments I found in similar post, including re-encoding, offsetting the audio, speeding up the audio (to match the video duration), adding audio filters (aresample=async with different values) and none of them helped - all I managed to achieve is getting a sync at the beginning of the video, but the audio keeps getting more and more delayed (at 50 mins the delay is like 15 seconds).
Any ideas what am I doing wrong? Any hints would be highly appreciated.
Thanks a lot in advance!