All Garmin HRM show elevated Heart rate when beginning a run

I can confirm same behaviour in my case.
Fenix 7X, HRM-Pro or HRM-Pro Plus.
I was thinking that it is a strap problem, bought new one and issue stil exist.
Bot not on every run, sometimes everything is fine, sometimes likes on your screens.
Some other observations, sometimes running dynamics start recording after couple second, the same for running power or breath frequency.

What I can say, I was using my HRM-Pro with my previous Fenix 6X and didn’t have such issues.

Use an ECG gell on the pads, that cured the problem with me. 

Only a guess - if it's cold weather and your skin is dry at the start of the run the chest strap may not activate straight away. So the OHR is used initially and I guess you're seeing cadence lock instead of actual HR. Then you begin to sweat and the strap wakes up and starts broadcasting.

No cadence lock on the bike, only running.

Not a chance, I have set for running power mode to „Jacket only” it is cold and I have my watch on the jacket sleeve.
And watch is giving me message „External HR connected” befor I start activity.

I don't really consider 40 degrees Fahrenheit to be cold where I am... but some may.  I have used ECG gel with the same results and do a warmup before starting my workout.  I use a Stryd foot pod so I  have have my HRMs set to heart rate only. I guess maybe cadence lock is a possibility, but i would think the ECG gel would combat that.  I purchase an polar h10 I'm going to test against.  

Maybe it is a moisture issue.. this is from a brick workout, the run was after the bike so i was plenty sweaty.  The heart rate looks fine. Maybe I'll try soaking the strap....

You might be right. 
In my situation I observed that I don’t have this effect when I do longer warmup before run. 
If I only stretch then effect return. 
Tomorrow should have ECG gel, will test it. 

Thank you for advice but in reality it doesn’t make sense to make whole strap wet, only electrodes need to be wet to conduct circuit better and yes I’m moisture them always before I use the strap. 
I was using HRM-Run I’m the past (with Fenix 3 and Fenix 6), two years ago I bought HRM-Pro and used it with Fenix 6 and to be honest I don’t remember to have issues like this. 

It started with Fenix 7 I bought year ago, first with my old HRM-Pro and now with new HRM-Pro+. 

I’m closer and closer to conclusion it is watch issue not the strap. Unfortunately I sold my Fenix 6 and cannot test these strap with it. 

Adam-1972 said:

in reality it doesn’t make sense to make whole strap wet, only electrodes need to be wet

Well since the electrodes do not hold water there is some sense in wetting the strap. Whether it makes sense to some or not, it works and it really is no hardship to do so, especially if it reduces/prevents problems.