I am planning on buying a cheap manual flash for outdoor photoshoots. But, I am not sure if those cheap non-TTL flashes can be triggered by any brand's non-TTL wireless trigger, or whether they require a specific one.
Provided that all you want a wireless solution to do is tell the flash when to fire, any transmitter/receiver set that are compatible with one another will be able to do that. The easiest way to insure this is to buy a set that includes both a transmitter and a receiver.
This is based on the assumption that the flash, camera, and triggering system in question all have hot shoe connectors with standard center pins. The center pin connector is near universal among most consumer cameras with external hot shoe connectors other than a few Sony/Minolta cameras that do not use the standard center pin arrangement.
All of the following hot shoe arrangements us a center pin for the "fire" command. All you need for the "fire" command is the connection pictured at the lower right. The other connection points in the other connectors are for more sophisticated communication between the camera and flash for a specific, proprietary system. But all of them will signal the "fire" command via the center pin.
Yes if you only care about manual sync, and are using manual (non-TTL) gear, and add-on radio triggers that attach to the foot or sync port of the flash, then brand should not matter. You can mix and match a LumoPro LP-180 with manual triggers from Phottix, Yongnuo, Godox, PocketWizard, RadioPopper, whatever. There's an ISO standard that enforces foot/hotshoe dimensions and the wiring/placement of the sync pin/contact. So this one signal is compatible across brands that adhere to it (the only exception is the older Sony/Minolta iISO hotshoe).
If, however, you are using a cheapie manual flash with a built-in radio receiver, you could use any brand of manual trigger to fire in sync, but only a same-brand/compatible transmitter lets you access additional features, such as remote power/zoom control (or even HSS): e.g., Yongnuo YN-660 and YN-560-TX; Godox TT600 and X1T; LumoPro LP180R and Phottix Odin/Strato transmitters.