It looks like you're trying to "parlay" equity to buy multiple homes. It doesn't work like that.
All "equity" is is what the house is worth (what you can sell it for) minus what you owe. The only way to use "equity" on one house to purchase another is to cash it out. That can be done by either selling the first house or by taking it out using some kind of loan.
If you sell it you'll get back what you sell the house for less what you owed on it (less any transaction costs). That could also be part of a linked transaction where you make the purchase of the second home contingent on the sale of the first.
You can also cash out equity by refinancing the first home or taking out a home equity loan. That reduces the equity on the first home by how much you take out in cash. You could then use it to make a down payment on a second home. Depending on how much equity you cash out and what your resulting loan-to-value and debt-to-income ratios are, banks may not be too keen on this approach.
Say you have 15,000$ equity in home #1, and use it to finance a down payment for home #2, does home #2 have $15,000 in initial equity?
That would mean that home #1 would then have zero equity (which the bank won't let you do) or that you've sold home #1 to cash out the equity.
if so, can that $15,000 be used to finance a down payment on a 3rd home?
Sure, but the same applies, either home #2 will have no equity or you have to sell it.
So equity can be moved from one home to another in certain circumstances, but it can't be duplicated.