The Great Pyramid of ASCII

Retina, 59 56 54 52 bytes

{`^( *)/.(.*)..
$1 /$2\¶$0
T`/\\`Ro`(?<=^ +/).+(?=.)

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Sample output

    /\
   /\\\
  /\///\
 ///\\\/\
//\\///\/\

Pyth - 27 26 bytes

Reduces by the operation given in the OP until it repeats, which is the case for the blank line.

j_.e+*kdb.ujXtPtPNK"\/")_K

Test Suite.

Jelly, 28 26 25 24 bytes

QṚ,QyḊḊṖṖj@QµÐĿµJ’⁶ẋ⁸żYṚ

-4 bytes thanks to Dennis

Recipe:

QṚ,QyḊḊṖṖj@QµÐĿµJ’⁶ẋ⁸żYṚ - one argument: input()
Q  Q       Q             - set of Left=input(): "/\"
 Ṛ                       - reverse Left: "\/"
  ,                      - Left-pair-Right: ["\/","/\"]
     ḊḊṖṖ                - dequeue Left twice, then pop twice: input()[2:-2]
    y                    - translate Right with mapping in Left: swaps internal slashes
         j@              - join Right with separator Left (@ swaps operands)
            µ  µ         - chain separators to form a 1,1,1 chain of chains
             ÐĿ          - loop while results are unique and collect them
                J        - yield [1,...,len(Left=input())]
                 ’       - decrement: [0,....len(input())-1]
                  ⁶      - " "
                   ẋ     - repeat Left Right times: ["", " ", ...]
                    ⁸ż   - zip Right and Left (⁸ is the link's Left argument):
                                ...pads the loop results
                      Y  - joins Left with line-feeds
                       Ṛ - reverse Left

(serve with lemonade, those pyramids make for thirsty workers)

Cook up your own slash pyramid at TryItOnline, or try out all the OP's suggested tasters