A Difficult Application of Modular Arithmetic

Looking to input a $p$ for the first line:

p=6; 

Build a list of primes from the 3rd to $p-1$:

plist=Table[Prime[n],{n,3,p-1}]; 

Then build a list as follows:

bmod=Outer[Mod,{3*Prime[p]},plist] 

Here's the code so far with the output:

p=6; plist=Table[Prime[n],{n,3,p-1}]; bmod=Outer[Mod,{3*Prime[p]},plist] {{4,4,6}} 

Now I'd like to multiply each element of the list by $2$, but keep them mod their original prime, so instead of:

{{8,8,12}} 

I get:

{{3,1,1}} 

Then, I'd like to add this list to the original, still keeping the modular bounds:

{{3,1,1}} + {{4,4,6}} = {{2,5,7}} 

And then repeat this process several times, always adding to the previous line:

{{4,4,6}} {{2,5,7}} {{0,6,8}} {{3,0,9}} etc. 

Assuming you've followed my so far, how would I do that?