You could define your rules this way:

In[226]:= Clear["Global`*"]

In[227]:= expr = m/n == p/q

Out[227]= m/n == p/q

In[228]:= % /. (x_/y_ == w_/z_) -> y z (x/y == w/z)

Out[228]= n q (m/n == p/q)

In[229]:= % /. (x_ (z_ == w_)) -> x z == x w

Out[229]= m q == n p

In[230]:= % /. x_ == y_ -> x - y == 0

Out[230]= -n p + m q == 0

Thanks Kay

This is OK for the case stritcly limited to m/n == p/q but m,n,p,q may represent much longer expressions. So unless I create four new variables, your solution does not work as in the example below :

expr = (Cos[Pi/4] Power[3 x , (4)^-1])/x^3  == Log[var]/( x y z )

One of the problem is that MMA will evaluate an expression unless held from it and a solution based on pattern matching to change any expression seems to me difficult to generalize.