If moving to a new tank is only because of the algae, I would not do it. You just need continuous export.
What do all algae (and cyano too) need to survive? Nutrients. What are nutrients? Ammonia/ammonium, nitrite, nitrate, phosphate and urea are the major ones. Which ones cause most of the algae in your tank? These same ones. Why can't you just remove these nutrients and eliminate all the algae in your tank? Because these nutrients are the result of the animals you keep.
So how do your animals "make" these nutrients? Well a large part the nutrients comes from pee (urea). Pee is very high in urea and ammonia, and these are a favorite food of algae and some bacteria. This is why your glass will always need cleaning; because the pee hits the glass before anything else, and algae on the glass consume the ammonia and urea immediately (using photosynthesis) and grow more. In the ocean and lakes, phytoplankton consume the ammonia and urea in open water, and seaweed consumed it in shallow areas, but in a tank you don't have enough space or volume for this, and, your other filters or animals often remove or kill the phytoplankton or seaweed anyway. So, the nutrients stay in your tank.
Then, the ammonia/ammonium hits your rocks, and the periphyton on the rocks consumes more ammonia and urea. Periphyton is both algae and micro-animals, and is the reason your rocks change color after a few weeks from when they were new. Then the ammonia goes inside the rock, or hits your sand, and bacteria there convert it into nitrite and nitrate. However, the nutrients are still in your tank.
Also let's not forget phosphate, which comes from solid organic food particles. When these particles are eaten by microbes and clean up crews, the organic phosphorus in them is converted into phosphate. However, the nutrients are still in your tank.
So whenever you have algae or cyano "problems", you simply have not exported enough nutrients out of your tank compared to how much you have been feeding (note: live rock can absorb phosphate for up to a year, making it seem like there was never a problem. Then after a year, there is a problem).
So just increase your nutrient exports. You could also reduce feeding, and this has the same effect, but it's certainly not fun when you want to feed your animals :)
--and--
What is Periphyton?
Periphyton is what turns your rocks different colors. You know... the white rocks you started with in saltwater, or the grey rocks (or brown wood) you started with in freshwater. After several months or years, the rocks become a variety of different colors and textures. Why? Because the periphyton that has grown on it is a mix of different living things, with different colors, and thicknesses. And the important part is: It is LIVING. And the thicker it is, the more living material there is.
That's right: The colored stuff that has coated your rocks is all living organisms. Sponges, microbes, algae, cyano, biofilms, and of course coralline in saltwater. After all, "peri" means "around the outside", and "phyto" means "plant". Have you ever slipped while walking on rocks in a stream? That's the periphyton that made it slippery. It can be a very thin coating on the rocks, sometimes paper thin, but it covers the entire surface, especially across the top which gets more light.
There is a lot of photosynthetic organisms in periphyton, and this means that they need light; but they need nutrients too (ammonia, nitrate, phosphate). And as you might guess, the organisms will prefer the illuminated portions of the rocks, and will grow to intercept food particles in the water, based on the water flow. Just think about how sponges orient their openings for water flow; the micro sponges in periphyton do it too, but on a tiny scale.
What about in caves and holes? Well periphyton here don't get much light, so they are primarily filter feeders (non-photo). So they REALLY grow and position themselves to be able to intercept food particles in the flowing water. You'll sometimes see little tree-like arms or branches sticking out to get the particles; these are usually "forams", otherwise known as foraminifera. They require a lot of food particles in the water. Think of them as coral polyps without the coral.
Reef studies have shown that at certain depths, more of the filtering of the water comes from periphyton and benthic algae, than comes from the phytoplankton which filters the deeper water. And in streams, almost all the filtering is done by periphyton because the water is so shallow. So, what you have on rocks that are "mature" or "established" is a well-developed layer of periphyton (and all the good natural things that comes from it, like nutrient absorption and food generation). They have nice colors too.
This is why mandarin fish can eat directly off the rocks of an "established" tank (tons of pods grow in and consume the periphyton), but not on the rocks of a new tank. Or why some animals can lay their eggs on established rocks, but not new ones. Or why established tanks seem to "yo-yo" less than new ones (the periphyton is a giant, self-adjusting filter). Even tangs can eat periphyton directly when it's thick enough. Yes periphyton can also develop on the sand, but since the sand is moved around so much, the periphyton does not get visibly thick like it does on rocks. So thick periphyton on established rocks is your friend. And totally natural too. That's why there are no pure white rocks in natural reefs. Keep in mind though I'm not referring to nuisance algae on rocks; I'm only referring to the layer of coloring and textures that coats the rocks, and the little arm-like structures that stick out from under rocks.
But what happens when you "scrub all the stuff off your rocks"? Well, you remove some of the periphyton, which means you remove some of your natural filter and food producer. What if you take the rocks out of the water and scrub them? Well now you not only remove more of your natural filter and food producer, but the air is going to kill even more of the microscopic sponges. And what if you bleach the rocks? Well, goodbye all filtering and food producing for another year. It's an instant reduction of the natural filtering that the periphyton was providing. So it's best to not do these things at all.
However, what if you just re-arrange the rocks? Well, some of the periphyton that was in the light, now will be in the dark; so this part will die. And some of the periphyton that was in the dark will now be in the light, so it will not be able to out-compete some photosynthetic growth and thus will be grown over and will partially die too. And even if the light is the same in the new location, the direction and amount of water flow (and food particles) will change; forams and micro sponges that were oriented to get food particles from one direction will now starve. So, since the light and food supply is cut off in the new location, the filtering that the periphyton was providing stops almost immediately, due only to your re-arranging of the rocks.
Starvation takes a little longer. The periphyton organisms won't die immediately, since they have some energy saved up; but instead they will wither away over several weeks. So on top of the instant reduction in filtering that you get by just moving the rocks, you get a somewhat stretched-out period of nutrients going back into the water. And after all this, it takes another long period of time for the periphyton to build up to the levels it was at before: 1 to 2 years. Even changing the direction of a powerhead will affect the food particle supply in the area it used to be pointed at. This is why "mature tanks" take 1 or 2 years to develop.
So a good idea is to try to keep everything the same. Pick your lighting, flow, layout, and then try to never again change anything. In other words, treat your rocks just like the rocks on a reef. It's a different way of thinking, but you should have a stronger natural filter and food producer because of it.
