What the OP is intending to do is progressively sell shares and re-buy (or transfer a trust) to trigger capital gains and subsequently increase the cost base of those shares so that if the whole portfolio was sold at a single point in time in the future there would be a minimal gain realised.
The OP would do the selling/transferring in a controlled manner over a number of years in order not to breach the threshold of paying tax ~$20k of income or if eligible for SAPTO then approx ~$30k of income.
Depending on the OPs circumstances they should explore (in consultation with a licenced Financial Planner) whether they would be better of contributing the shares to super as a non-concessional contribution.
