I do a bit of mapping, and have done extensive research on how to represent elevation on a map, because I also use such features extensively.
One of the best ways to do so is through contours, explained in this wiki which is a handy read if you want a broad overview on the subject. Contours can be effectively used on a battle map as a way to represent elevation, as they're the official cartography tools for this very purpose. Despite the fact that contours are generally used to show slopes and depressions on a map, this very selfsame tool can be used for the purpose of showing cliffs and sinkholes if you know what you're doing and label the lines as far as elevation appropriately. This is appropriate for battle maps because using contour lines you can use the battle map itself as a tool to show elevation. As a rule of thumb, the closer the lines are together, the steeper the slope. In order to show contours which describe the features you're asking to showcase, another website here reads as follows
Cliffs are the easiest to identify on a topological map. When looking for a cliff on a map, look for multiple contour lines merging into one line. The last contour line with ticks on it facing towards the ground is a cliff.
An Example of such a feature can be found here:
Another website addresses sinkholes specifically here
Sinkholes and other depressions are marked by closed contour lines with hatch marks. Not always perfectly round, some sinkholes may be elongated such as the one in the upper right corner. Other sinkholes may swallow streams or be filled with water such as the one to the right of "Hollow".
Piggybacking on this general contouring information, in order to use this idea on a pre-gridded battle map, in a game use the contour markings to represent sinkholes and cliffs, all you need to do is simply "expand" a section that you wish to show as a section that the PCs and monsters are traveling through. You may need to tell the players that the grid's scale has changed in order to show the appropriate amount of detail. Depending on the size of the battlemap and the scale you're using, you may need to reorient the matt, erase the markings, draw the next section to continue the chase.
Using the grid and the markings, you should be able to place the minis to show how far up or down the players and monsters are, simply add the minis next to one or more die, with each dice being whatever scale you decide. d10s works perfectly, as you have a one foot granular control in lots of 10. Maybe even color code them for the various minis to make it simpler to tell which is which. Or color code the dice for the elevation itself. For instance: red 10 feet, orange 20 feet, yellow 30 feet, green 40 feet, blue 50 feet, purple 60 feet, white 70 feet, brown 80 feet, grey 90 feet, black 100 feet, and combining the colors to get the exact number of feet needed.
Further expanding on this idea, you could pipe cleaners (preferably something colorful so that it shows up well against the map) to depict where the vines are, bent so they stand up by themselves. The spacing of the vines will also determine if the players can get from one to another.
I've also seen minis that have a base that has numbers on them that can be "dialed" to show HP. If these are used, instead of tracking HP, they could be used to show elevation in 10s of feet. Combine with a single d10 for the granularity you need.