Over the past decade, polar codes have received significant attraction and have been selected as the coding method for the control channel in fifth-generation (5G) wireless communication systems. However, conventional polar codes are reliant solely on binary ($2 \times 2$) kernels, which restricts their block length to being only powers of 2. In response, multi-kernel (MK) polar codes have been proposed as a viable solution to achieve increased flexibility in code length. This article proposes unrolled and pipelined architectures for encoding both systematic and nonsystematic MK polar codes, capable of high-throughput encoding of codes constructed with binary, ternary ($3 \times 3$), or binary-ternary mixed kernels. Furthermore, two novel nonsystematic stage-folded encoders, designed to minimize resource usage, have been introduced for the encoding of pure-ternary and MK codes. The proposed MK encoders additionally provide the functionality of dynamic kernel assignment. The proposed architectures exhibit an unprecedented level of flexibility by supporting 83 different codes and offering various architectures that provide tradeoffs between throughput and resource consumption. The FPGA implementation results demonstrate that a partially pipelined polar encoder of size$N=4096$operating at a frequency of 270 MHz gives a throughput of 1080 Gb/s. In addition, a new compiler scripted in Python is introduced to automatically generate HDL modules for the desired encoders. By inserting the desired parameters, a designer can simply obtain all the necessary VHDL files for FPGA implementation.