So the heart of early American conscious cultivation (bad-dum-sch!) of horticulture is in the city of Philadelphia. John Bartram (1699-1777) is generally recognized as the founder of American botany/horticulture/plant science/what-have-you. His estate on the shores of the Schuylkill River (opposite and <5 miles from center city Philadelphia), began in 1728. As a Quaker, he was deeply interested in plants as a manifestation of the divine -- as someone who was kicked out of his Quaker church (but attended anyway because eff them) he built his own house of stone and did not recognize the divinity of Jesus. ( Also paid his freed African American servants, but loathed native Americans for killing his father -- inconsistent humanitarianism. Fun facts about a fun man.) He was fascinated by the plants of the new world, and sought to incorporate them into European/Linneaus taxonomic systems. He went on specimen collecting and documenting expeditions to the Blue Ridge Mountains (1738), the Lake Ontario region (1751), the Catskills (1755), and the Carolinas (1760). He wrote books about each journey that you can click through to HERE. His efforts were widely applauded; in 1765 he was named Botanist to King George III -- in part thanks to Benjamin Franklin ardently lobbying on his behalf.

Anyway, where this connects to your question is that perhaps Bartram's greatest contribution was his facilitation of the trade in plants between the old world and the new; through his influence, wealthy colonists (especially fellow Quakers in the Philadelphia area, for whatever reason) became interested in amassing geographically diverse collections of plants, not only from across the new American continent, but also from around the world wherever other British botantists were collecting seeds. Meanwhile, British royals rapidly embraced American plants, in their gardening, especially Kalmias, Rhododendrons, and Magnolias, which Bartram not only collected and helped identify, but hybridized his own varieties of. Peter Collinson, a British merchant, enjoyed a close friendship with Bartram and helped him ship plants from the Americas, which evolved into a full nursery business continued by 3 generations of Bartrams. Bartram was the primary supplier, for instance, to Robert James Petre, 8th Baron Petre of Thorndon Hall, Essex, who was the foremost collector of North American trees and shrubs in Europe in the 1730s.

To get to the physical component of your question, Bartram became famous for his 'Bartram's boxes,' which Collinson would deliver to London, and distribute the contents to their owners there. (The tradition of boxes continued through several more generations of Bartram nurserymen/botantists.) Seeds, cuttings, rhizomes, and misc. zoological would all be packed in -- sometimes soil and sometime sawdust and sometimes sand and sometimes moss as necessary for maintaining the right humidity and temperature. (Very rarely, scrap paper, which was very expensive!) Still, obviously many plants and specimens would die in transit, even though sometimes they would go so far as to ask for the boxes to be stores underneath officer's bunks on the ship to ensure the least risk of water damage, etc. The boxes were 3' x 2.5' and wooden, in case that level of detail appears.

SOURCES: http://www.smithsonianmag.com/history/the-story-of-bartrams-garden-13572809/?no-ist

http://www.woodturningcenter.org/ChallengeVIII.html (has a nice contemporary engraving of such boxes)

Hazel Le Rougetel, "Philip Miller/John Bartram Botanical Exchange" Garden History Vol. 14, No. 1 (Spring, 1986) , pp. 32-39

Philip J Pauly, Fruits and Plains: The Horticultural Transformation of America (Cambridge, Mass.: Harvard University Press, 2007) -- Pauly is much more concerned with the ideological/conceptual movement of plants than the physical/material but may still be useful for you.

Misc knowledge from my head.

PS- Everyone interested in the least bit in any of this and who is near Philadelphia should absolutely still visit Bartram's Gardens. It's the oldest botanic garden in the US, still operational, and features a ginkgo tree which is the oldest in America -- because it is the last alive of the three original specimens brought to the continent from China in 1785!!! Also, they do awesome food justice work (the garden is near some public housing projects), which is a provocative and fruitful (ba-dum-schh!) way to think about incorporating an appreciation of our history into our visions for more just futures: https://bartramsgarden.org

With the settlement of Australia the First Fleet brought only seeds to start cultivation. This was likely due to convenience as it would take up less space but also because they were interested in seeing not only if crops they were aware of could grow in Australia but the way in which they would grow. Looking at sources like diary of Joseph Gidley King, Lt. Governor of Norfolk Island, we can see detailed records of the which crops proved successful not just in terms of could they grow, but the quality of the produce. We also see that this experimentation meant a huge variety of seeds were brought over not just wheat but also seeds for maize, peach trees, orange trees, banannas and guavas to name but a few. The one possible example of live plants being transported I can find is the listing of oak and myrtle trees on the list of cargo of a First Fleet ship. However, I'm afraid I can't find any comments on whether these were cuttings, live trees carefully stored or seeds.