Weak mixing in lattice models is informally the property that ``information does not propagate inside a system''. Strong mixing is the property that ``information does not propagate inside and on the boundary of a system''. In dimension two, the boundary of reasonable systems is one dimensional, so information should not be able to propagate there. This led to the conjecture that in 2D, weak mixing implies strong mixing. The question was investigated in several previous works, and proof of this conjecture is available in the case of finite range Gibbsian specifications, and in the case of nearest-neighbour FK percolation (under some restrictions). The present work gives a new proof of these results, extends the family of models for which the implication holds, and, most interestingly, provides a ``percolative picture'' of the information propagation.