I tried your flasher circuit and all it did was cook the transistors. So I took the liberty of redesigning the circuit to one that I tested and indeed works and it won't cook the transistor.I added notes but feel free to make any changes you like for different effects.I was to lazy to diagnose the problem with the first circuit.
This circuit has a problem too, in that when the NPN turns on there is no limit on the current from 9V+, through the LED, through the PNP emitter-base junction, through the NPN collector-emitter path, to ground. Perhaps there isn't enough base current into the NPN to drive things hard enough to cause problems, but I would like to seen some current limiting, probably at the PNP base.
The current to the LED is limited by the 330-470 ohm resistor.Because the LED has a forward voltage of 1.7V the current it draws is I= (9-1.7)/330= 22ma ,Although that seems high the actual is lower because the battery is not at full potential (8.5V) . That puts the current at around 20mA . Since the LED is on a pulse ,short bursts of slightly higher current isn't hurting the LED .This has been measured and confirmed .The 470ohm resistor would be better limiting the LED to around 15.5mA .This is also the reason that no limiting resistor is require at the base of the PNP transistor.Because only 20mA is being drawn across the transistor the current at the base is very small.About 0.5mA has been measured at most.
It's not the current through the LED that I was commenting on, but the current through the PNP emitter-base junction and NPN collector-emitter path. When the PNP is in saturation, that base current has little to do with the collector current, but is limited only by how much current that NPN transistor is pulling. The NPN transistor also has no current limit other than the base current multiplied by the current gain since there are no resistors in the PNP/NPN chain. Hey, it probably works fine in practice, but since we're critiquing circuits here I thought I would mention this issue.
I used LTSpice to test this (with 2n3904 NPN and 2n3906 PNP), just to see how much of an issue this actually is. I had to increase the "off delay" resistor to 20M before it would oscillate, but it did oscillate. As expected, the NPN collector current is influenced by the "flash rate" resistor value, but setting that at 10K gives me 100mA spikes. That's a lot of base current for that PNP, but probably not damaging. That current is also flowing through the LED, since that 330 Ohm resistor is not in that path. Once the NPN comes out of saturation then the 330 resistor comes into play.
I changed the "flash rate" resistor to 50k and the PNP base current spike was reduced to around 25mA. The LED current spiked up to 42 mA then stabilized at 16 mA during the flash.
I added a 1K resistor between NPN collector and PNP base and then the currents were as you might expect: PNP base current 4 mA, and LED current 16-21 mA.
No big deal, I just prefer to not have these type of currents dependent on transistor parameters.
Since I can't figure out how to embed your schematic, here's a link to it:
https://www.eevblog.com/forum/beginners/transistors-204336/?action=dlattach;attach=811830;image