Hi Obelix! With regards to the subject, alternation & moderation.
And regards to the actual issue (this is my first post since 2011, so bare with me), I have had a similar issue which as far as I can recall may have also been related to tricep pushdowns. For me, what it felt like was a tendon being pulled over a bone tightly, it was painful and a sort of clicking was felt.
It also got worse over time and carried over into other tricep exercises (possibly after enough time had passed to cause inflammation), until I started paying more attention to the comfort I felt in my positioning and movements.
I had the same issue with my knees and found that I would be trying to mimic what I saw when watching videos etc, such as foot position when performing squats, instead of adjusting to what felt natural and comfortable. Joint pain is a good indication that something is not right.
Although, what I would suggest is having your elbow looked at by a professional in case it needs rest or rehab, what I did was to drop weight and really slow down my movements. Pay attention to the point in the movement that hurts and try re-adjust your position (this could be the space between your hands, the flaring of your elbows etc.). Once you find what works, then take it slow and add weight over time while maintaining a healthy technique and making sure you aren't experiencing any unexpected discomfort.
You might find that you may have been favouring that side and also substituting form for heavier weight.
Also, for your tricep pushdowns, are you using rope or a bar? If you are using a bar, try rope.
Look after yourself<3
