Picard discusses differential equations with fixed singularities in the end of Chapter V (pp. 291-300). The special case of Painleve VI that he discovered is
written in the end on p. 299. Example of a second order equation whose solutions
have movable essential singularities is given on p. 291. He does not care to write the equation explicitly but only writes its solution. On the next page he gives a 3-d order equation, which he correctly credits to Jacobi; it has much worse, non-isolated movable singularities.
It is not a big deal to construct an example with a movable essential singularity, anyone can construct such examples:
$$(yy''-(y')^2)^2+4(y')^3y=0$$
has a family of solutions $\exp(1/(z-c))$ with essential singularities at $c$.
Whole Chapter V in Picard discusses singularities of differential equations, but this is a difficult reading for a modern mathematician, and the main reason is not the French.
Remarks. 1. Modern terminology is somewhat different from Picard's: in his example,
the function is not single-valued, and in the modern times one usually
applies the word "essential singularity" only to single-valued functions.
(My example above is single-valued).
It is interesting that R. Fuchs, Painleve and Gambier who much later discovered Painleve VI, do not mention Picard. Painleve was Picard's student (see, for example Math Genealogy), and he missed "Painleve VI" in his original classification of these equations.
I recommend the books by E. L. Ince, Ordinary differential equations in English and V. V. Golubev, Lectures on the analytic theory of ordinary differential equations in Russian and German.
