Visualize long addition with ASCII art
Pyth, 59 58 bytes

L.[dJhl`eSQ`b:jk_.u/+NsYT.t_MjRTQ00\0djbyMPQXyeQ0\+*J\-ysQ


Try it online. Test suite.

Way too long. Should golf more.

Explanation

L                  helper function y = lambda b:
        eSQ          largest number in input
      l`             length as string
     h               increment
    J                save to J
 .[d       `b        pad argument with spaces to that length

                             carry row:
                jRTQ           each input to base 10
              _M               reverse each result
            .t      0          transpose, padding with zeroes
    .u               0         cumulative reduce from 0:
         sY                      sum digits of column
       +N                        add previous carry
      /    T                     floor-divide by 10
   _                           reverse
 jk                            join by ""
:                     \0d      replace 0 by space

          number rows:
    PQ      all input numbers but last one
  yM        pad to correct length
jb          print on separate lines

           last number row:
  eQ         last input number
 y           pad to correct length
X   0\+      change first char to +

        separator row:
 J        width of input (saved in helper)
* \-      that many dashes

       result row:
 sQ      sum of inputs
y        pad to correct length