Python 2, 287 271 270 273 bytes

import time,random
r,n=input()
c=n*-~r or-r
while c:
 c-=1;L=[list(' '*i+'\\'+'  '*(r-i)+'/')for i in range(r)];x=c%-~r;time.sleep(.1);y=x+1;exec"L[x][y]=random.choice('-/\_'[L[x][y-1]in'\_'::2][y==2*r-x:]);y+=1;"*2*(r-x)
 for l in['']*99+L+[' '*r+'\xaf'*2]:print''.join(l)

Try it online!

Does not clear the screen on tio, but works in a console.

Gif of it running: