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how can I have equivalent exposure and identical white balance while changing f stops using speedlight? im shooting from shoes when it comes to the moment of changing f stops from 5 to 16 in order to get more depth of field, the white balance changes... any help would be greatful, thanks in advance.

Let’s pull the question apart...

Assuming a full strobe lit scene...

how to maintain exposure while changing f stop?

The only other variables in play are ISO and strobe power. So if you go from f/4 to f/8, you’d need to correspondingly either raise your ISO two stops or double your strobe power, twice.

how to maintain white balance while changing strobe power

Strobes naturally have a different white balance through the power range. The best thing you can do is use customs white balance and redo it after changing power output.

So, putting it together, to maintain exposure and white balance, you’d need to increase ISO or strobe power, or both, and then redo your white balance setting.

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Assuming a mixed lighting scenario...

how to maintain exposure while changing f stop

Let’s assume that you’re mixing natural and artificial light and are using the strobe as fill. If you go from f/4 to f/11, then your strobe must get 2^3 times more powerful to get the same amount of fill flash off the subject and into the camera.

But, also keep in mind that you’ve dropped your ambient capture as well. So you’ll have to lengthen the shutter speed by 3 stops as well.

how to maintain white balance

Well, if you’re mixing strobe and ambient, you’ve probably already placed a gel or filter over the strobe to balance it to your ambient. Not much should need to change. If your strobe swings so wildly at different power outputs, you may need to change filters/gels.

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A note on the inverse square law

In both scenarios, increasing strobe power is a solution for getting correct subject exposure when decreasing your f stop. Keep in mind that this will spill more light into your background. It may be necessary to instead mess with not only the lights power but also it’s placement distance from the subject.

If you are shooting with a TTL flash, and if you change aperture or ISO, the TTL exposure metering will recompute a different power level to give the same exposure. That is how TTL works. (Same exposure assumes if still within the capability of the power level, if the flash can still do it).

With a manual flash, you have to adjust the power level yourself, but either way, the power level changes.

But the white balance of flash units changes with power level, which seems to be your complaint. Higher power level is greater amps of current though the flash tube, which changes ionization level of the Xenon gas, which changes spectrum somewhat. That is just how the physics is.

There are a couple of studio flash models with an internal computer that tries to balance volts and amps to maintain a relatively equal white balance with power level. But except for that couple of studio lights, the huge majority of all flash units change white balance with power level. Buying one of those couple would be the only answer to your question. The Paul C. Buff Einstein model would be the most affordable of these. Profoto has the other.

NOT intending to imply any great numerical precision here, but :

Most studio lights use lower voltage for lower power, and then very roughly perhaps 200 degrees K more red from full power to 1/32 power.

Speedlights always use full voltage, but cut off the flash duration very abruptly for low power, and then very roughly perhaps 400 degrees K more blue from full power to 1/32 power. A few studio lights use this same method.

There is a very easy solution for white balance. Just add a white balance card into the scene of the first test shot, and use a good white balance tool (preferably in raw software) to click the card in that test photo. That tells the computer "this spot is neutral color, so make it be neutral color". Neutral color means equal red, green, blue components, so no color cast. That works great, and is trivially easy, very fast, and very good.

My site has as sample of the speedlight color change, on second page at
http://www.scantips.com/lights/whitebalance.html

TL;DR. Won't be exact/identical from shot to shot, but gel your flash to match the ambient, shoot RAW, and compensate for aperture changes with ISO or consider using TTL if you don't care if the ambient exposure level shifts.

The white balance is likely changing because you're getting different amounts of ambient light in, and the ambient light color doesn't match your flash's color. This is going to be a problem for you no matter what white balance you have set in-camera. In this situation, gelling the flash to match the color of the ambient light, so a single white balance setting will still work for the image is useful. As would shooting RAW, instead of jpeg, so you can "reset" the white balance after the fact in post.

However, keep in mind that speedlights often aren't color consistent across all power settings, so changing the flash power may change its white balance. And that going over sync speed will rob you of flash power, d

Exposure-wise, if you rapidly swap aperture settings, you must also compensate with a change in ISO setting if you want to maintain the same flash/ambient exposure.

Ambient exposure is controlled by ISO, aperture, and shutter speed.

Flash exposure is controlled by ISO, aperture, flash power, and flash-to-subject distance.

So, say, -3.3EV on aperture (f/5→f/16) also means going +3.3EV (10x) on the ISO, so ISO 100→1000. If you compensated with a combination of ISO and shutter speed (say, ISO 100→400 + shutter speed of 1/250→1/80), then the shutter speed portion (+1.3EV) also has to be compensated for by flash power (say, 1/4→1/2+1/3EV) because shutter speeds below sync speed don't affect flash exposure.

Or, if you don't care about the ambient exposure shifting with the aperture, you could just use TTL to maintain the flash exposure level through ISO and aperture changes.

Call it a speed light or a strobe or electronic flash (a rose by any other name smells just as sweet). These flash units output a bright blitz, short in duration. Because the flash happens super quickly, we can create a stop-motion setup. If done correctly, we can freeze the action of athletes or even freeze hummingbird wings.

As to changing the lens aperture for different effects, under conditions of continuous light, we can freely change the shutter speed / aperture combination to get different results. When electronic flash is the principal light source, changing these combinations are limited. This is because the short flash duration confines us to use shutter speeds that allow synchronization of the shutter with the flash.

If we are not mindful to use only those shutter speeds that harmonize with the flash, we are likely to get a frame that sees the flash incompletely. We call this a synchronization error. Most cameras are very finicky and we can only use 1/125 of a second or slower. Because different shutters sync at different speeds, best you check your camera manual to see this limitation for yourself.

That being said it is not likely that you can change your aperture from f/5 to f/16 at you whim. Such a change requires a shutter speed adjustment to compensate for the change in the exposing light action. If we are working under continuous lighting then we can do this at will. With a flash as main light, other considerations are afoot.

Likely you are working under ambient light. When the shutter speed is set slow, you are allowing the flash and the ambient light to comingle and this act complicates white balance. You flash is rich with blue and likely the ambient is rich with a rosy hue.

When flash and ambient mingle, the outcome is likely uncertain. What I am saying is, the skill to white balance under mixed lighting must be acquired by study plus a bit of luck won’t hurt.